On the Geometry and Analysis of Graphs

Transcription

1 On the Geometry and Analysis of Graphs Habilitationsschrift vorgelegt am der Fakultät für Mathematik und Informatik der Friedrich-Schiller-Universität Jena von Dr. Matthias Keller geboren am in Karl-Marx-Stadt, jetzt Chemnitz

2 Gutachter: Prof. Dr. Alexander Grigor yan, Universität Bielefeld Prof. Dr. Jürgen Jost, Max-Planck-Institut Leipzig Prof. Dr. Daniel Lenz, Friedrich-Schiller-Universität Jena Prof. Dr. Karl-Theodor Sturm, Universität Bonn Vorgelegt am: Erteilung der Lehrbefähigung am:

3 Acknowledgements The submission of a habilitation thesis poses a landmark in academic life which puts me in the privileged position to look back, sum up and, first and foremost, to express my gratitude to the various people that shared an active part during this time. The first acknowledgement is due to Daniel Lenz who supported me as an advisor, teacher, coauthor and friend during these years. His continuous support, inspiration, guidance and encouragement are a corner stone of the present work. Furthermore, this work would not have been possible without the expertise of my further coauthors Frank Bauer, Michel Bonnefont, Sylvain Golénia, Sebastian Haeseler, Bobo Hua, Xueping Huang, Jun Masamune, Norbert Peyerimhoff and Radoslaw Wojciechowski. This research was pursued at the Friedrich Schiller University Jena and the Hebrew University Jerusalem. In Jena I am indebted to the very productive, friendly and joyful atmosphere in the analysis group which would not be the same without my dear colleagues and friends Siegfried Beckus, Felix Pogorzelski and last but not least Marcel Schmidt. In Jerusalem, I was blessed with the welcoming, open and vibrant environment of the Hebrew University and, in particular, the generous hospitality of my hosts Jonathan Breuer, who I gladly count among my coauthors, and Dan Mangoubi. During the past years I had the luxury to enjoy the hospitality of various institutions and colleagues. In particular, I am glad to thank Alexander Grigor yan (Bielefeld University), Jürgen Jost (MPI Leipzig), Wolfgang Woess (Graz University of Technology), Peter Stollmann and Ivan Veselic (Chemnitz University of Technology), Jozef Dodziuk (Graduate Center CUNY, NYC), Alexander Teplayev and Michael Hinz (University of Connecticut), Shing-Tung Yau and Gabor Lippner (Harvard), Balint Virag (University of Toronto) and Nabila Torki-Hamza (Université de Carthage) for the invitations and the profitable and pleasant time. I was supported in this research by various grants and institutions. I gratefully acknowledge the support of the German Science foundation for the project Geometry of discrete spaces and spectral theory of non-local operators, the Golda Meir Fellowship, the Israel Science Foundation (grant No. 1105/10 and No. 225/10) and BSF grant No I am further thankful that Birkhäuser, Springer, the London 1

4 2 ACKNOWLEDGEMENTS Mathematical Society, Elsevier and Mathematical modeling of natural phenomena granted their permission to reproduce the original articles within the thesis. My parents, sisters and friends were always a source of strong support during these years and I am thankful that I always found an open ear. My life would not be half as exciting without my wonderful children Elliott, Lumen and Ferris who I cherish for being cheerful at day time and sleeping peacefully at night time. Finally, the love and support of my wife Yvonne can not be measured or expressed in words and I am endlessly grateful for the life we are sharing.

5 Contents Acknowledgements 1 Structure of the work 7 Part 1. Introduction 9 Synopsis 11 Notation 15 Chapter 1. Dirichlet forms on discrete spaces Graphs and Laplacians The heat equation Uniformly positive measure Weakly spherically symmetric graphs Sparseness 31 Chapter 2. Intrinsic metrics Definition and basic facts Liouville theorems Domain of the generators and essential selfadjointness Isoperimetric constants and lower spectral bounds Volume growth and upper spectral bounds Volume growth and l p -independence of the spectrum 51 Chapter 3. Curvature on planar tessellations Set up and definitions Curvature and the bottom of the spectrum Decreasing curvature and discrete spectrum The l p spectrum Curvature on planar graphs 63 Bibliography 67 Part 2. Original Manuscripts 73 Chapter 4. M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, Journal für die reine und angewandte Mathematik 2012 (2012),

6 4 CONTENTS Chapter 5. Chapter 6. Chapter 7. Chapter 8. Chapter 9. Chapter 10. Chapter 11. Chapter 12. Chapter 13. Chapter 14. Chapter 15. M. Keller, D. Lenz, Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation, Mathematical modeling of natural phenomena: Spectral Problems 5 (2010), M. Keller, D. Lenz, R. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Mathematische Zeitschrift 274 (2013), M. Bonnefont, S. Golénia, M. Keller, Eigenvalue asymptotics for Schrödinger operators on sparse graphs, arxiv: B. Hua, M. Keller, Harmonic functions of general graph Laplacians, Calculus of Variations and Partial Differential Equations 51 (2014), X. Huang, M. Keller, J. Masamune, R. Wojciechowski, A note on self-adjoint extensions of the Laplacian on weighted graphs, Journal of Functional Analysis 265 (2013), F. Bauer, M. Keller, R. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, to appear in Journal of the European Mathematical Society. 239 S. Haeseler, M. Keller, R. Wojciechowski, Volume growth and bounds for the essential spectrum for Dirichlet forms, Journal of the London Mathematical Society 88 (2013), F. Bauer, B. Hua, M. Keller, On the l p spectrum of Laplacians on graphs, Advances in Mathematics 248 (2013), M. Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Mathematische Annalen 346 (2010), M. Keller, N. Peyerimhoff, Cheeger constants, growth and spectrum of locally tessellating planar graphs, Mathematische Zeitschrift 268 (2011), M. Keller, Curvature, geometry and spectral properties of planar graphs, Discrete & Computational Geometry 46 (2011),

7 CONTENTS 5 Zusammenfassung in deutscher Sprache 357 Lebenslauf 361 Schriftenverzeichnis 371 Ehrenwörtliche Erklärung 375

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9 Structure of the work The present work is divided into two parts. In the introductory part, the main results of this work are presented; the second part consists of original manuscripts. The overall theme is the connection of geometry, analysis and probability on discrete spaces modeled by Dirichlet forms. The geometric concepts under investigation are distance and curvature and the derived consequences range from Liouville theorems, essential selfadjointness to spectral theory of selfadjoint operators. The first part is divided into three chapters: 1. Dirichlet forms on discrete spaces [KL12, KL10, KLW13, BGK13]. 2. Intrinsic metrics [HK14, HKMW13, BKW14, HKW13, BHK13]. 3. Curvature on planar tessellations [Kel10, KP11, Kel11] (and also [BGK13, BHK13]). The exposition of the first and the second chapter will be partially published in a slightly modified form in the survey article [Kel14b] and the last chapter in the survey article [Kel14a]. The references listed after the topics of the chapters above refer to the original manuscripts that form the second part of this thesis. Theses references are listed below: [KL12] M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, Journal für die reine und angewandte Mathematik 2012 (2012), [KL10] M. Keller, D. Lenz, Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation, Mathematical modeling of natural phenomena: Spectral Problems 5 (2010), [KLW13] M. Keller, D. Lenz, R. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Mathematische Zeitschrift 274 (2013), [BGK13] M. Bonnefont, S. Golénia, M. Keller, Eigenvalue asymptotics for Schrödinger operators on sparse graphs, arxiv: [HK14] B. Hua, M. Keller, Harmonic functions of general graph Laplacians, Calculus of Variations and Partial Differential Equations 51 (2014),

10 8 STRUCTURE OF THE WORK [HKMW13] X. Huang, M. Keller, J. Masamune, R. Wojciechowski, A note on self-adjoint extensions of the Laplacian on weighted graphs, Journal of Functional Analysis 265 (2013), [BKW14] F. Bauer, M. Keller, R. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, to appear in Journal of the European Mathematical Society. [HKW13] S. Haeseler, M. Keller, R. Wojciechowski, Volume growth and bounds for the essential spectrum for Dirichlet forms, Journal of the London Mathematical Society 88 (2013), [BHK13] F. Bauer, B. Hua, M. Keller, On the l p spectrum of Laplacians on graphs, Advances in Mathematics 248 (2013), [Kel10] M. Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Mathematische Annalen 346 (2010), [KP11] M. Keller, N. Peyerimhoff, Cheeger constants, growth and spectrum of locally tessellating planar graphs, Mathematische Zeitschrift 268 (2011), [Kel11] M. Keller, Curvature, geometry and spectral properties of planar graphs, Discrete & Computational Geometry 46 (2011),

11 Part 1 Introduction

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13 Synopsis The impact of the geometry on the spectral and stochastic features of Laplacians and their semigroups is studied in many areas of mathematics. Indeed, Laplacians on Riemannian manifolds and graphs share a lot of common elements. Despite of this, various geometric notions such as distance and curvature which arise canonically in the Riemannian framework have no immediate analogue in the discrete setting. For distances it even turns out that the naive approach to define a distance via the combinatorial graph distance leads to serious disparities in the comparison of the theory of discrete and continuum models. The development and investigation of suitable notions of distance and curvature is a major theme of this work. The guiding perspective is that Laplacians on Riemannian manifolds and on graphs both originate from so called Dirichlet forms. From a physical perspective, such quadratic forms may intuitively be understood to model an energy functional. In our presentation, we focus on Dirichlet forms on discrete spaces. These spaces have the virtue that they often allow for a very explicit and rather non-technical treatment. Nevertheless, our presentation is designed to pave the way for a treatment in the general case. In the first chapter, we introduce the set up and basic concepts. In particular, we present a one-to-one correspondence between (weighted) graphs and regular Dirichlet forms on a discrete space. We introduce Laplacians and their semigroups via these forms, discuss their basic properties and give examples. Next, we take a look at the heat equation and a property called stochastic completeness which serves as a characterization for uniqueness of bounded solutions to the heat equation. The original treatment of these two sections is found in the original works [KL12, KL10] attached in the second part. The chapter is completed by the discussion of three classes of examples. The first class consists of graphs over certain measure spaces, namely, the measure of vertices is assumed to be bounded below by a positive constant. The results here are based on [KL12, BHK13]. Secondly, we consider graphs with a weak form of spherical symmetry and study various spectral and probabilistic properties [KLW13]. Finally, we take a closer look at sparse graphs based on results obtained in [BHK13, KL10]. 11

14 12 SYNOPSIS In the second chapter, we consider a notion of distance and corresponding notions such as volume. Here, the naive approach to define a distance for graphs by a version of the combinatorial graph distance leads to disparities compared to Riemannian manifolds. This was first observed by Wojciechowski [Woj08, Woj11], see also [CdVTHT11a, KLW13]. In particular, these disparities appear when one considers unbounded operators on graphs. On the other hand, in the case of the normalized graph Laplacian which is always a bounded operator the combinatorial graph distance provides the proper analogue to the Riemannian case. Therefore, the combinatorial graph distance is, in some sense, the natural metric for the normalized Laplacian while it is unsuitable for general operators. This suggests that one should look for an appropriate notion of distance for a given operator. This approach proved to be very effective in the context of strongly local regular Dirichlet forms. There, so called intrinsic metrics were used to extend various results from Riemannian geometry to a very general framework, see the systematic pioneering investigation of such metrics by Sturm [Stu94]. Recently, a concept of intrinsic metric was introduced for general regular Dirichlet forms by Frank/Lenz/Wingert [FLW14] (which circulated as a preprint 5 years prior to its publication). Here, we use such metrics to study operators on general graphs. In this way, we obtain results which seem to be the natural discrete analogues to the Riemannian setting. As a highlight of this chapter we point out a Cheeger inequality from [BKW14]. This comes, in some sense, as a surprise since, at first glimpse, it is not clear how a notion of distance enters the definition of an isoperimetric constant. Indeed, this solves an open problem of Dodziuk/Kendall [DK86] from Next to this result, Liouville theorems of Yau and Karp, Gaffney s theorem for essential selfadjointness, Brooks and Sturm s upper bounds for the bottom of the (essential) spectrum and Sturm s p-independence of the spectra are proven for graphs, see [HK14, HKMW13, HKW13, BHK13] for the original manuscripts. All of these results are the first in this direction for general graphs. They all contain the normalized Laplacian as a special case and sometimes improve the result known for this case. In the third and final chapter, we address a notion of curvature. We restrict ourselves to planar graphs with standard weights and consider a very intuitive geometric notion of curvature going back to ideas of Descartes. Our main focus lies on spectral consequences of upper curvature bounds. We give lower and upper bounds on the bottom of the spectrum in terms of curvature as they were obtained in [KP11]. Furthermore, we characterize discreteness of the spectrum in terms of curvature. This is an analogue to a theorem of Donnelly/Li and is found in [Kel10]. In this case, we can determine the first order of

15 SYNOPSIS 13 the eigenvalue asymptotics which is an application of [BGK13]. Finally, we apply the results of [BHK13] to discuss p-independence of the spectrum.

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17 Notation We list the most important notation of this thesis to serve as a small index. The page number refers to the place where these objects are defined. X... a discrete infinitely countable space (page 17). (b, c)... a (weighted) graph with edge weight b and killing term c (page 17). m... a measure on X (page 18). n... the normalizing measure for a graph on X which is the sum of the edge weights about a vertex and the killing term (page 18). Deg... the weighted vertex degree which is the sum of the edge weights about a vertex divided by the measure (page 36). deg... the combinatorial vertex degree which is the number of edges emanating from a vertex (page 18). Q on D... the generalized form associated to a graph (page 18). Q... the regular Dirichlet associated to a graph (page 21). L on F... the generalized Laplacian associated to a graph (page 19). L... the selfadjoint operator associated to Q which is a restriction of L (page 21).... the Laplacian on a graph with standard weights and the counting measure (page 23). n... the normalized Laplacian on a graph with standard weights and the normalizing measure (page 23). 15

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19 CHAPTER 1 Dirichlet forms on discrete spaces This chapter is dedicated to setting the stage and introducing the basic notions and concepts. We start by defining weighted graphs on a discrete measure space and show that there is a one-to-one correspondence to regular Dirichlet forms on this space. Via these forms, we obtain selfadjoint operators on the corresponding l 2 space by general theory and characterize basic features such as boundedness and the fact that compactly supported functions are in the domain of these operators. These operators give rise to a semigroup which can be extended to the l p spaces for all 1 p. We discuss that their generators are restrictions of a general Laplacian. Secondly, we discuss the heat equation for bounded functions. Uniqueness of solutions can be characterized by a property called stochastic completeness at infinity. Furthermore, the stability of this property under embeddings into supergraphs is discussed. Finally, we discuss classes of examples of graphs which allow for a more specific investigation of certain aspects. First, we consider graphs over a measure space whose measure allows for a positive lower bound on singelton sets. Secondly, we study graphs with a weak spherically symmetry and, thirdly, we look at graphs with relatively few edges which we refer to as sparse graphs. The first three sections summarize the results of the original manuscript [KL12] and also some of the material in [KL10, BHK13]. The first section also appeared as an introductory section in the survey article [Kel14b]. The fourth section presents the results of [KLW13] and the fifth section is mainly taken from [BGK13] and from [KL10] in one instance Graphs and Laplacians Graphs. Let X be a discrete and countably infinite space. A graph (b, c) over X is a symmetric function b : X X [0, ) with zero diagonal and b(x, y) <, x X, y X and c : X [0, ). We say two vertices x, y X are neighbors if b(x, y) > 0 and we write x y. We can think of b(x, y) as the bond strength which increases with the strength of the interaction between 17

20 18 1. DIRICHLET FORMS ON DISCRETE SPACES x and y. The function c can be thought to describe one-way-edges to a virtual point at infinity or as a potential or as a killing term. If c 0, then we speak of b as a graph over X. We call a sequence (x k ) of pairwise distinct vertices a path if x k x k+1 for all indices k. We say a graph is connected if for all x, y X there is a finite path with x and y as end vertices. If a graph is not connected we may restrict our attention to the connected components. Therefore, we assume in the following that the graphs under consideration are connected. A measure of full support on X is given by a function m : X (0, ) which is extended additively to sets via m(a) = x A m(x), A X. If we fix a measure m, then we speak of a graph (b, c) or b over (X, m). Given a graph (b, c), an important special case of a measure is the normalizing measure n defined as n(x) = y X b(x, y) + c(x), x X. Another important special case is the counting measure m 1. We say a graph is locally finite if every vertex has only finitely many neighbors, that is, if the combinatorial vertex degree deg is finite at every vertex: deg(x) = #{y X x y} <, x X. We say a graph has standard weights if b : X X {0, 1} and c 0. In this case, the normalizing measure n equals the combinatorial vertex degree deg. Obviously, by the summability assumption on b, graphs with standard weights are locally finite General forms and Laplacians. In this subsection, we introduce the forms and Laplacians on rather large spaces. Later, we restrict these objects to spaces with rather more structure. We let C(X) be the set of real valued functions on X and C c (X) be the subspace of functions in C(X) of finite support The general form. For a graph (b, c) over X, the general quadratic form Q : C(X) [0, ] is given by with domain Q(f) = 1 2 x,y X b(x, y) f(x) f(y) 2 + x X D = {f C(X) Q(f) < }. c(x) f(x) 2

21 1.1. GRAPHS AND LAPLACIANS 19 Since Q 1 2 is a semi norm and satisfies the parallelogram identity, by polarization, Q yields a semi scalar product on D via Q(f, g) = 1 b(x, y)(f(x) f(y))(g(x) g(y)) + c(x)f(x)g(x). 2 x,y X x X in The general Laplacian and Green s formula. For functions F = {f C(X) y X b(x, y) f(y) 2 < for all x X}, we define the general Laplacian L : F C(X) by Lf(x) = 1 b(x, y)(f(x) f(y)) + c(x) m(x) m(x) f(x). y X Obviously, we have F = C(X) in the locally finite case and, in general, C c (X) F. Next, we come to a Green s formula which was first shown in [HK11], see also [HKLW12]. Lemma 1.1 (Green s formula, Lemma 4.7 in [HK11]). For f F and ϕ C c (X) 1 b(x, y)(f(x) f(y))(ϕ(x) ϕ(y)) + c(x)f(x)ϕ(x) 2 x X x,y X = x X Lf(x)ϕ(x)m(x) = x X f(x)lϕ(x)m(x) Solutions and harmonic functions. An important tool to study various analytic and probabilistic properties of graphs are solutions to certain equations. Here, we briefly introduce solutions to elliptic equations. Later, in Section 1.2, we consider also solutions to the heat equation. A function f F is called a solution (respectively, subsolution or supersolution) for λ R if (L λ)f = 0 (respectively, (L λ)f 0 or (L λ)f 0). A solution (respectively, subsolution or supersolution) for λ = 0 is is said to be harmonic (respectively, subharmonic or superharmonic). We say a function f C(X) is positive if f(x) 0, x X, and non-trivial and strictly positive if f(x) > 0, x X. A Riesz space is a linear space equipped with a partial ordering which is compatible with addition, scalar multiplication and where the maximum and the minimum of two functions exist. Immediate examples are C(X), C c (X), F, D, and the canonical l p -spaces, 1 p, introduced below. An important fact that is needed in the subsequent material is the following. In order to study existence of non-constant (respectively,

22 20 1. DIRICHLET FORMS ON DISCRETE SPACES non-zero) solutions for λ 0 in a Riesz space, it suffices to study positive subharmonic functions. The well-known lemma below follows from the fact that the positive and negative parts and the modulus of a solution to λ 0 are non-negative subharmonic functions. Lemma 1.2. Let (b, c) be a connected graph over (X, m) and F 0 F be a Riesz space. If there are no non-constant positive subharmonic functions in F 0, then there are no non-constant solutions for λ 0 in F 0 and, in particular, any solution for λ < 0 is zero Dirichlet forms and their generators. The forms and Laplacians introduced above are defined on spaces without a norm. Next, we will consider restrictions of these objects to Hilbert and Banach spaces. Let l p (X, m) be the canonical real valued l p -spaces, p [1, ], with norms ( f p = f(x) p m(x) x X ) 1 p f = sup f(x). x X, p [1, ), As l (X, m) does not depend on m, we also write l (X). For p = 2, we have a Hilbert space l 2 (X, m) with scalar product f, g = x X f(x)g(x)m(x), f, g l 2 (X, m), and we denote the norm by = 2. For the domain of the general Laplacian F, one always has l (X) F. The inclusion of l p (X, m), p [1, ), in F does not hold in general, but holds in the case of uniformly positive measure, that is, inf m(x) > 0. x X Dirichlet forms. In our context, a Dirichlet form is a closed quadratic form q with domain D l 2 (X, m) such that, for f D, we have 0 f 1 D and q(0 f 1) q(f), where g h denotes the maximum of g and h and g h denotes the minimum of g and h. A form q is called regular if D C c (X) is dense in D with respect to q = (q( ) + 2 ) 1 2 and in C c (X) with respect to. We denote the restriction of Q to D(Q (N) ) = D l 2 (X, m)

23 1.1. GRAPHS AND LAPLACIANS 21 by Q (N), where the superscript (N) indicates Neumann boundary conditions. By Fatou s lemma, Q (N) can be seen to be lower semicontinuous and, thus, closed. It follows directly that Q (N) is a Dirichlet form. Moreover, closedness of Q (N) yields immediately that the restriction of Q to C c (X) is closable. We define Q = Q (D) by Q(f) = 1 2 x,y X D(Q) = C c (X) Q, b(x, y) f(x) f(y) 2 + x X c(x) f(x) 2, f D(Q). Here, the superscript (D) indicates Dirichlet boundary conditions. It can be seen that Q is a Dirichlet form (see [FOT11, Theorem 3.1.1] for a proof in the general setting and, for a proof of this fact in the graph setting, see [Sch12, Proposition 2.10]). Obviously, Q is regular. As it turns out, by [KL12, Theorem 7], all regular Dirichlet forms on (X, m) are given in this way a fact which can be also derived from the Beurling-Deny representation formula [FOT11, Theorem and Theorem 5.2.1]. Theorem 1.3 (Theorem 7 in [KL12]). If q is a regular Dirichlet form on l 2 (X, m), then there is a graph (b, c) such that q = Q (D) Markovian semigroups and their generators. By general theory (see e.g. [Wei80, Satz 4.14]), Q yields a positive selfadjoint operator L with domain D(L) viz Q(f, g) = L 1 2 f, L 1 2 g, f, g D(Q). By the second Beurling-Deny criterion, L gives rise to a Markovian semigroup e tl, t > 0, which extends consistently to all l p (X, m), p [1, ], and is strongly continuous for p [1, ). Markovian means that for functions 0 f 1, one has 0 e tl f 1. We denote the generators of e tl on l p (X, m), p [1, ), by L p, that is, D(L p ) = { f l p 1 (X, m) g = lim t 0 t (I e tl )f exists in l p (X, m) } L p f = g and L is defined as the adjoint of L 1. It is a direct consequence from the Green s formula that L 2 is a restriction of L. Moreover, in [KL12], it is also shown that L p, p [1, ], are restrictions of L. Theorem 1.4 (Theorem 5 in [KL12]). Let (b, c) be a graph over (X, m) and p [1, ]. Then, L p f = Lf, f D(L p ).

24 22 1. DIRICHLET FORMS ON DISCRETE SPACES Boundedness of the operators. We next comment on the boundedness of the form Q and the operator L. The theorem below is taken from [HKLW12] and an earlier version can be found in [KL10, Theorem 11]. Theorem 1.5 (Theorem 9.3 in [HKLW12]). Let (b, c) be a graph over (X, m). Then the following are equivalent: ( ) (i) X [0, ), x 1 m(x) y X b(x, y) + c(x) is a bounded function. (ii) Q and, in particular Q, is bounded on l 2 (X, m). (iii) L and, in particular L p, is bounded for some p [1, ]. (iv) L and, in particular L p, is bounded for all p [1, ]. Specifically, if the function in (i) is bounded by D <, then Q 2D and L p 2D, p [1, ] The compactly supported functions as a core. It shall be observed that C c (X) is, in general, not included in D(L). Indeed, one can give a characterization of this situation. The proof is rather immediate and we refer to [KL12, Proposition 3.3] or [GKS12, Lemma 2.7] for a reference. Lemma 1.6. Let (b, c) be a graph over (X, m). Then the following are equivalent: (i) C c (X) D(L). (ii) LC c (X) l 2 (X, m) (iii) For each x X, the function X [0, ), y 1 b(x, y) is m(y) in l 2 (X, m). In particular, the above assumptions are satisfied if the graph is locally finite or inf m(y) > 0, x X. y x Moreover, either of the above assumptions implies l 2 (X, m) F. Proof. The equivalence of (ii) and (iii) follows from the abstract definition of the domain of L. The equivalence of (i) and (ii) is a direct calculation, see [KL12, Proposition 3.3] and the in particular statements are also immediate, see [KL12, GKS12] Graphs with standard weights. In this subsection, we consider two important special cases. We say a graph has standard weights if b : X X {0, 1} and c 0. For these graphs, we consider two measures which play a prominent role in the literature.

25 1.2. THE HEAT EQUATION 23 For the counting measure m 1, we denote the arising operator L on l 2 (X) = l 2 (X, 1) by. It operates as f(x) = y x(f(x) f(y)), f D( ), x X. By Lemma 1.5, the operator is bounded if and only if deg is bounded. By Lemma 1.6, we still have C c (X) D( ) in the unbounded case since standard weights imply local finiteness. For the normalizing measure n = deg, we call the arising operator L on l 2 (X, deg) the normalized Laplacian and denote it by n. The operator n acts as 1 n f(x) = (f(x) f(y)), f l 2 (X, deg), x X, deg(x) y x and, by Lemma 1.5, the operator n is bounded by The heat equation In this section, we discuss the heat equation on l (X). A function u : [0, ) l (X), t u t that is continuous on [0, ) and differentiable on (0, ) at every x X is called a solution to the heat equation with initial condition f l (X) if Lu t = t u t, t > 0, u 0 = f. Continuity for t Lu t (x), x X, on [0, ) can easily be seen and the validity of the heat equation extends to t = Stochastic completeness. In the case c 0, uniqueness of solutions to heat equation in l (X) can be characterized by a property that is called stochastic completeness (or conservativeness or honesty or non-explosion, depending on the context). This property deals with the question of whether the semigroup leaves the constant function 1 invariant. There is a huge body of literature from various mathematical fields investigating this property, so for references, we restrict ourselves to mentioning the work for discrete Markov processes in the late 50 s by Feller [Fel58, Fel57] and Reuter [Reu57], for manifolds the work of Azencott [Aze74] and of Grigor yan [Gri88, Gri99], for positive contraction semigroups the work of Arlotti/Banasiak [AB04] and of Mokhtar-Kharroubi/Voigt [MKV10] and, for strongly local Dirichlet forms, the work of Sturm [Stu94]. A graph is called stochastically complete if e tl 1 = 1 for some (all) t > 0. There is a physical interpretation of this property. This concerns the question of whether heat leaves the graph in finite

26 24 1. DIRICHLET FORMS ON DISCRETE SPACES time. Assume the graph is not stochastically complete, i.e., e tl 1 < 1 for some t > 0. Let 0 f l 1 (X, m) model the distribution of heat in the graph at time t = 0. Then, the distribution of heat at time t > 0 is given by e tl f and the amount of heat at time t > 0 is given by f(x)m(x), x X e tl f(x)m(x) = e tl f, 1 = f, e tl 1 < f, 1 = x X where the right hand side is the amount of heat at time t = 0. Hence, the amount of heat in the graph decreases in time, if the graph is not stochastically complete Stochastic completeness at infinity. Usually, stochastic completeness is studied for forms with vanishing killing term since a non-vanishing killing term immediately results in the loss of heat. Here, we consider all regular Dirichlet forms on (X, m), including nonvanishing killing term and study a property called stochastic completeness at infinity. So, to deal with a non-vanishing c, we have to replace e tl 1 by the function t ( M t (x) = e tl 1(x) + e sl c ) (x)ds, x X. m 0 In [KL12] it is shown that M t is well defined, satisfies 0 M t 1, and, for each x V, the function t M t (x) is continuous and even differentiable. In the special case c 0, we obtain M t = e tl 1, whereas for c 0 we obtain M t > e tl 1 for connected graphs. By the interpretation above, the term e tl 1 can be seen to be the amount of heat contained within the graph at time t and the integral can be interpreted as the amount of heat killed within the graph up to the time t by the killing term. Thus, M t can be interpreted as the amount of heat, which has not been transported to the boundary of the graph at time t. The following theorem is one of the main results of [KL12]. For related results in the case c 0 and m 1 see [Fel58, Fel57, Reu57, Woj08]. Theorem 1.7. (Theorem 1 in [KL12]) Let (b, c) be a graph over (X, m). Then, for any λ < 0, the function w := 0 λe tλ (1 M t )dt satisfies 0 w 1, solves (L λ)w = 0, and is the largest non-negative l 1 with (L λ)l 0. In particular, the following assertions are equivalent: (i) M t 1 for some (all) t > 0. (ii) The function w is nontrivial.

27 1.2. THE HEAT EQUATION 25 (iii) For any (some) λ < 0, there is no non-trivial u l (X) such that Lu = λu. (iv) For any (some) f l (X) there is a unique solution U : [0, ) l (X), t u t to Lu = t u, u 0 = f. Definition 1.8. We say a graph is stochastically complete at infinity if one of the equivalent assertions of the theorem above is fulfilled. A further equivalence by an weak Omori-Yau-Principle can be found in [Hua11b] for c Subgraphs. Next, we discuss stochastic completeness (at infinity) from the perspective of a graph having subgraphs with this property. It is well known from the theory of random walks that a graph is transient whenever it has a transient subgraph. Such a statement is wrong for stochastic completeness and stochastic completeness at infinity. First, we present a result that shows that a graph can be stochastically completed at infinity by adding a killing term. Theorem 1.9 (Theorem 2 in [KL12]). For any graph (b, c) over (X, m), there is c : X [0, ) such that (b, c + c ) is stochastically complete at infinity. The idea behind the proof given in [KL12] is that through the additional killing term so much heat is already killed in the graph that no heat reaches the boundary in finite time. Secondly, we present a result that shows the phenomena discussed above. Namely, we can complete a graph by embedding the graph into a larger supergraph. A subgraph (b W, c W ) of a graph (b, c) over (X, m) is given by a subset W of X and the restriction b W of b to W W and the restriction c W of c to W. The graph (b, c) is then called a supergraph to (b W, c W ). Given a measure m on X we denote its restriction to W by m W. The subgraph (b W, c W ) then gives rise to a form Q W on the closure of C c (W ) in l 2 (W, m W ) with respect to Q with associated operator L W. Theorem 1.10 (Theorem 3 in [KL12]). Any graph is the subgraph of a graph that is stochastically complete at infinity. This supergraph can be chosen to have a vanishing killing term if the original graph has a vanishing killing term. The idea of the proof in [KL12] is to attach sufficiently many stochastically complete graphs to each vertex. For example, one may choose line graphs or simply single edges.

28 26 1. DIRICHLET FORMS ON DISCRETE SPACES Given the two results above, it seems desirable to give a sufficient condition for a graph not being stochastically complete at infinity in terms of subgraphs. To this end, we introduce subgraphs with Dirichlet boundary conditions. Given a graph (b, c) over (X, m) and W X, the subgraph with Dirichlet boundary conditions (b (D) W, c(d) W ) over (W, m W ) is given by where d W (x) := y X\W above, we get a form Q (D) W b (D) W = b W and c (D) W = c W + d W, b(x, y), x W. Analogous to the definition on l2 (X, m) and an operator L (D) W. With this terminology we get a result that complements the theorem above. Theorem 1.11 (Theorem 4 in [KL12]). A graph is not stochastically complete at infinity, whenever it has a subgraph with Dirichlet boundary conditions that is not stochastically complete at infinity Uniformly positive measure In this section, we discuss a class of graphs whose measure can be bounded by a positive constant from below. It seems that these results have no direct analogues in the non-discrete setting. We first show a Liouville theorem. As a consequence, we obtain a criterion for essential selfadjointness, equality of the Dirichlet and Neumann form and an explicit description of the domain of the l p generators. These results are taken from [KL12]. Secondly, we discuss a spectral inclusion result for the spectrum σ(l 2 ) of the l 2 generator L 2 in the spectrum σ(l p ) of the l p generators L p which is taken from [BHK13]. The condition below is on the measure space (X, m) only. We say the measure m is uniformly positive if (M) inf x X m(x) > 0. For example, this holds if m is constant as in the case of the counting measure or deg. For some results, we may weaken (M) to a condition that additionally takes into account the combinatorial structure of a graph over (X, m): (M*) n=1 m(x n) = for all infinite paths (x n ) A Liouville theorem. The following theorem is a slightly stronger statement than [KL12, Lemma 3.2]. The argument of the proof is the same as in [KL12] and, as it is very simple, we include it here. Theorem Let (b, c) be a connected graph over (X, m) satisfying (M*). Then, any positive subharmonic function in l p (X, m), p [1, ), is zero.

29 1.3. UNIFORMLY POSITIVE MEASURE 27 Proof. Let f be positive and subharmonic. Then, Lf(x) 0 evaluated at some x gives 1 f(x) y X b(x, y) b(x, y)f(y) Thus, whenever there is x x with f(x ) < f(x) there must be y x such that f(x) < f(y). By connectedness, such x and x exist whenever f is non-constant. Letting x 0 = x, x 1 = y and proceeding inductively, there is a sequence (x n ) of vertices such that 0 f(x) < f(x n ) < f(x n+1 ), n 0, since we assumed that f is positive. Now, (M*) implies that f is not in l p (X, m), p [1, ). On the other hand, if f is constant, then (M*) again implies f 0. y X Domain of the generators. We discuss an application of the above theorem to determine the domain of the generator on l p. The other essential ingredient of the proof is that L p is a restriction of L. Theorem 1.13 (Theorem 5 in [KL12]). Let (b, c) be a graph over (X, m) such that every infinite path has infinite measure, (M*). Then, D(L p ) = {f l p (X, m) Lf l p (X, m)} for all p [1, ) Uniqueness of the form and essential selfadjointness. In the l 2 case, we get the following theorem that shows that the form with Dirichlet and Neumann boundary conditions coincide and we get a result on essential selfadjointness. Theorem 1.14 (Theorem 6 in [KL12]). Let (b, c) be a graph over (X, m) such that every infinite path has infinite measure (M*). Then, Q (D) = Q (N). If, additionally, LC c (X) l 2 (X, m), then C c (X) D(L) and the restriction of L to C c (X) is essentially selfadjoint. Remark. Recall that if the graph has uniformly positive measure (M), this both implies (M*) and also LC c (X) l 2 (X, m) by Lemma 1.6. Let us comment on the history of essential selfadjointness results for graph Laplacians. For standard weights and the counting measure such a result was first shown by Wojciechowski [Woj08]. The first correct proof in the general case is found in [KL12]. Later, somewhat weaker results were obtained by [JP11] and, independently, by [TH10]. Results of this type involving magnetic Schrödinger operators were later proven in [Gol14, GKS12].

30 28 1. DIRICHLET FORMS ON DISCRETE SPACES Spectral inclusion. In this section we discuss the spectral inclusion σ(l 2 ) σ(l p ) under the assumption of uniformly positive measure. This result was proven in [BHK13]. Theorem 1.15 (Theorem 2 in [BHK13]). Let (b, c) be a graph over (X, m) with uniformly positive measure (M). Then, for any p [1, ], σ(l 2 ) σ(l p ). The basic observation for the proof of the theorem is that (M) implies l p (X, m) l q (X, m), 1 p q, and, thus, D(L p ) D(L q ), 1 p q <, as under the assumption (M), we can determine D(L p ) explicitly by Theorem The abstract reason behind the result above is that, in the case of uniformly positive measure, the semigroup e tl is ultracontractive. In Section 2.6 we show that even an equality holds under a certain volume growth assumption (even without the assumption of uniform positivity of the measure). In general the inclusion is strict. For example for a regular tree with standard weights, the bottom of the l 2 spectrum of the normalized Laplacian is known to be positive. On the other hand, the normalized Laplacian is bounded and, thus, the constant function 1 is in the domain of the l generator and an eigenfunction to the eigenvalue 0. Hence, the l spectrum and by duality also the l 1 spectrum contains 0 which is not in the l 2 spectrum. (Of course, the same argumentation is true for the Laplacian with standard weights and the counting measure.) 1.4. Weakly spherically symmetric graphs The next class we consider are graphs which have a weak spherical symmetry. Often, symmetry is defined via existence of certain automorphisms. Our notion is much weaker, namely, that the graphs allow for an ordering into spheres such that certain curvature type quantities are spherically symmetric. For such graphs we show that the heat kernel is a spherically symmetric function, give a criterion for pure discrete spectrum, bounds for the spectral gap and present a characterization for stochastic completeness at infinity. The results presented here are originally proven in [KLW13]. We fix a vertex o X which we call the root and consider spheres and balls r S r = S r (o) = {x X d(x, o) = r} and B r = B r (o) = S i (o) about o of radius r and S 1 =. Here, d(x, y) is the combinatorial graph distance, that is, the minimal number of edges in a path connecting x and y. Define the outer and inner curvatures k ± : X [0, ) i=0

31 1.4. WEAKLY SPHERICALLY SYMMETRIC GRAPHS 29 by and let k ± (x) = 1 m(x) y S r±1 b(x, y), x S r, r 0, v = c m. We refer to k ± as curvatures since Ld(o, ) = k ( ) k + ( ) is referred to as a curvature-type quantity, such as mean curvature, see [DK88, Hua11a, Web10]. We call a function f : X R spherically symmetric if its values depend only on the distance to the root o, i.e., if f(x) = g(r), x S r (o), for some function g defined on N 0. Definition A graph (b, c) over (X, m) is called weakly spherically symmetric if there is a vertex o such that k ± and v are spherically symmetric functions Symmetry of the heat kernel. We start by discussing the consequences of weak spherical symmetry for the associated heat kernel. For the operator L we know, by the discreteness of the underlying space, that there exists a map p : [0, ) X X R, which we call the heat kernel associated to L, with e tl f(x) = y X p t (x, y)f(y)m(y), for all f l 2 (X, m), x X, t > 0. For a locally finite graph denote the averaging operator A : l 2 (X, m) l 2 (X, m) by Af(x) = 1 f(y)m(y). m(s r ) y S r We say that the heat kernel p is spherically symmetric if e tl commutes with A for all t > 0. In this case p t (o, ) is a spherically symmetric function for all t > 0. Spherical symmetry of p can be characterized by the graph being weakly spherically symmetric. Theorem 1.17 (Theorem 1 in [KLW13]). A graph (b, c) over (X, m) is weakly spherically symmetric if and only if the heat kernel is spherically symmetric. In [KLW13] also a heat kernel comparison theorem is proven involving the curvatures k ±.

32 30 1. DIRICHLET FORMS ON DISCRETE SPACES Spectral gap. In this section we discuss an estimate on the spectral gap for weakly spherically symmetric graphs. The spectral gap is given by the bottom of the spectrum σ(l) of L, that is, λ 0 (L) = inf σ(l). The spectrum is said to be purely discrete if the spectrum consists only of eigenvalues of finite multiplicity that have no accumulation point. The geometric quantity we use to estimate λ 0 (L) from below involves the volume of balls as well as the measure of the boundary of balls. For a set W X, we define the boundary W of W as the set of edges leaving W, i.e., W = {(x, y) W X \ W x y}. The map b can be considered as a measure on the edges and for sets U X X we write b(u) = b(x, y) and, in particular, b( W ) = (x,y) U (x,y) W b(x, y). Theorem 1.18 (Theorem 3 in [KLW13]). Let (b, c) be a weakly spherically symmetric graph over (X, m). If m(b r ) a = b( B r ) <, then r=0 λ 0 (L) 1 a. Moreover, the spectrum is purely discrete. The proof uses an Allegretto-Piepenbrink type theorem as it was proven in [HK11]. This theorem states that the bottom of the spectrum λ 0 can be characterized by the existence of positive supersolutions for λ λ 0. Such positive supersolutions are presented for λ < 1/a. To show discreteness of spectrum we present positive supersolution outside of balls B r. This yields that the bottom of the spectrum of the restricted operators is larger than the inverse of the sum in the theorem above starting from r + 1. This yields that these bottoms converge to infinity. As the restrictions are finite rank perturbations of the original operator, they share the same essential spectrum. Thus, the essential spectrum of L must be empty. In [KLW13] there is also a comparison theorem for the spectral gap involving the curvature quantities k ±. The proof of the comparison

33 1.5. SPARSENESS 31 theorem uses the heat kernel comparison and a discrete version of a socalled theorem of Li [KLVW13] which extracts λ 0 from p t by taking a limit Stochastic completeness at infinity. In this subsection we present a characterization of stochastic completeness at infinity for weakly spherically symmetric graphs by the divergence of a sum similar to the one above in Theorem We may also consider c as a measure on the vertices, i.e., c(w ) = x W c(x), W X. Theorem 1.19 (Theorem 5 in [KLW13]). A weakly spherically symmetric graph (b, c) over (X, m) is stochastically complete at infinity if and only if m(b r ) + c(b r ) =. b( B(r)) r= Sparseness In this section, we discuss a class of graphs with relatively few edges, a property which we refer to as sparseness. The results presented here are based on [BGK13]. There they are proven for Schrödinger operators on graphs with standard weights, which are Laplacians minus a potential. However, for general graphs the proofs carry over verbatim. In [BGK13] a hierarchy of notions of sparseness is introduced. The most general notion involves the so-called (a, k)-sparse graphs. Stronger notions are almost sparse graphs and then sparse graphs. For (a, k)-sparse graphs, the number of edges in a finite set are few compared to the boundary edges and the vertices of the set, where a is the ratio for the boundary edges and k for the vertices of the set. For almost sparse graphs a can be chosen to be arbitrary small at the expense of larger k. For sparse graphs a can be chosen to be zero. That is, the number of edges are few with respect to the number of vertices only. This is discussed in detail below. There is a close relationship to graphs which satisfy a strong isoperimetric inequality. These are graphs where the number of edges in a finite set are few with respect to the number of boundary edges. In fact, these are (a, k)-sparse graphs where k can be chosen to be zero. For all (a, k)-sparse graphs one can determine the form domain, characterize discreteness of the spectrum and prove eigenvalue asymptotics. These asymptotics are even better in the case of almost sparse graphs. For sparse graphs and graphs which satisfy a strong isoperimetric inequality, we then also discuss estimates for the bottom of the spectrum which are sharp in the case of regular trees.

34 32 1. DIRICHLET FORMS ON DISCRETE SPACES Notions of sparseness. Let (b, c) be a graph over (X, m). We start with the most general notion of sparseness which includes the other notions as special cases. A graph is called (a, k)-sparse for a, k 0, if for all finite W X b(w W ) a ( b( W ) + c(w ) ) + km(w ). In the case of standard weights the inequality reads as 2#E W a# W + k#w, where E W are the edges with starting and ending vertex in W. This case is treated in [BGK13]. There non-positive c are allowed as well. A graph is called almost sparse if for all ε > 0 there is k ε 0 such that the graph is (ε, k ε )-sparse. Finally, sparse graphs are such graphs where a can be chosen to be zero, i.e., a graph is called sparse or k-sparse if it is (0, k)-sparse. For graphs with standard weights, the assumption of k-sparseness reads as 2#E W k#w. The well known concept of an isoperimetric inequality is a special case of (a, k)-sparseness. A graph is said to satisfy a strong isoperimetric inequality if there is α such that n(w ) α(b( W ) + c(w )), where n is the normalizing measure n(x) = y b(x, y) + c(x), x X. In particular, it can be seen by direct calculation that a graph satisfies an isoperimetric inequality with α > 0 if and only if it is (a, 0)-sparse with a = (1 α)/α Characterization of the form domain. In this section we characterize the form domain of Q to be a certain l 2 space by (a, k)- sparseness of the graph. Furthermore, we characterize purely discrete spectrum of L in this case and present estimates for the eigenvalue asymptotics. Every function f on X induces a form on C c (X) l 2 (X, m) by pointwise multiplication. Given a form q on C c (X) we write f q on C c (X), if ϕ, fϕ q(ϕ, ϕ) for all ϕ C c (X). This will be used below for the function f = (1 ã)n/m + k with certain constants ã and k. For a function f : X R, we define f = sup inf f(x). K X finite x X\K In the case (n/m) =, we enumerate the vertices X = {x j } j 0 such that (n/m)(x j ) (n/m)(x j+1 ), j 0. Moreover, if L has purely discrete spectrum, then we enumerate the eigenvalues λ j (L), j 0, of L in increasing order and counting multiplicity.

35 1.5. SPARSENESS 33 Theorem 1.20 (Theorem 2.2 in [BGK13]). Let (b, c) be a graph over (X, m). Then the following are equivalent: (i) The graph is (a, k)-sparse for some a, k 0. (ii) For some ã (0, 1) and k 0 (1 ã)(n/m) k Q (1 + ã)(n/m) + k on C c (X). (iii) For some ã (0, 1) and k 0 (iv) D(Q) = l 2 (X, n). (1 ã)(n/m) k Q on C c (X). In this case, L has pure discrete spectrum if and only if (n/m) =. Furthermore, (1 ã) lim inf j λ j (L) (n/m)(x j ) lim sup j λ j (L) (n/m)(x j ) (1 + ã) In [BGK13] it is discussed how the constants a, k and ã, k can be estimated against each other. The most involved step in the proof of the equivalence is the implication (i) (ii) which uses isoperimetric techniques. The implication (ii) (iv) is obvious. Furthermore, the implication (iv) (iii) follows immediately from the closed graph theorem and the implication (iii) (ii) is a consequence of Kato s inequality and algebraic manipulation. The remaining statements are essentially a consequence of (ii) and the Min-Max principle Almost sparseness and eigenvalue asymptotics. For almost sparse graphs we get even better eigenvalue asymptotics. Theorem 1.21 (Theorem 3.2 in [BGK13]). Let (b, c) be an almost sparse graph over (X, m). Then for every ε > 0 there is k ε 0 such that on C c (X) (1 ε)(n/m) k ε Q (1 + ε)(n/m) + k ε. Then D(Q) = l 2 (X, n). Furthermore, L has pure discrete spectrum if and only if (n/m) =. Moreover, in this case lim j λ j (L) (n/m)(x j ) = 1. The only related results for graphs that we are aware of are found in [Moh13] for the adjacency matrix on sparse finite graphs. The proof follows from Theorem 1.20 and the explicit control of the constants ã, k in terms of a, k.

36 34 1. DIRICHLET FORMS ON DISCRETE SPACES Sparseness and the bottom of the spectrum. For a function f : X [0, ), we define f 0 = inf sup K X finite x X\K f(x). As sparse graphs are a special case of almost sparse graphs, we have D(Q) = l 2 (X, n) and the same estimate for the eigenvalue asymptotics. Moreover, we get better estimates for the bottom of the spectrum. Theorem 1.22 (Theorem 1.1 in [BGK13]). Let (b, c) be a k-sparse graph over (X, m). Then, for any ε (0, 1), (1 ε) n m k ( ) 1 2 ε ε Q (1 + ε) n m + k ( ) 1 2 ε ε, on C c (X). Furthermore, and (n/m) 0 2 lim j k 2 λ j (L) (n/m)(x j ) = 1. ( (n/m) 0 k ) λ 0 (L). 2 It can be seen that the estimate above is sharp in the case of regular trees with standard weights Strong isoperimetry and the bottom of the spectrum. In this subsection, we consider consequences of strong isoperimetric inequalities. From [KL10, Proposition 14] the next theorem follows immediately. The form inequality in the theorem below should be compared to the inequality in the three theorems above. Theorem 1.23 (Proposition 14 in [KL10]). Let (b, c) be a graph over (X, m) that satisfies a strong isoperimetric inequality with parameter α > 0. Then (1 1 α 2 )(n/m) Q (1 + 1 α 2 )(n/m), on C c (X). In particular, (1 1 α 2 )(n/m) 0 λ 0 (L). Again the estimate above is sharp in the case of regular trees with standard weights.

37 CHAPTER 2 Intrinsic metrics This chapter is dedicated to study the consequences of geometric notions related to distance. The starting point of the investigation is the realization that the combinatorial graph distance is not suitable in the case of unbounded operators. In particular, if one considers volume criteria for stochastic completeness, it was observed by Wojciechowski in his Ph.D. thesis [Woj08] that the criteria obtained for graphs differ significantly from corresponding results in the case of manifolds. Further results in this direction were observed in [CdVTHT11a, KLW13, Woj11]. As remedy, so-called intrinsic metrics can be used. This is the theme of this chapter. For strongly local Dirichlet forms, intrinsic metrics have been shown to be very effective to study various topics, see [Stu94]. Recently, this concept was generalized to all regular Dirichlet forms. It was first systematically studied and applied effectively by Frank/Lenz/Wingert in [FLW14], (see also [MU11] for an earlier mentioning of the criterion for certain non-local forms). By the virtue of these metrics various results can be shown for general graphs for the first time. From this perspective earlier results for the normalized Laplacian appear as a special instance. The relevance of intrinsic metrics stems from the fact that they provide the existence of suitable cut-off function. This allows for the application of methods known from manifolds and partial differential equations. While applying these methods one faces the challenge of the absence of a pointwise Leibniz rule and as a consequence the absence of a chain rule. In some cases the mean value theorem serves as a first step in the right direction, however, to obtain sharp results stronger estimates are of the essence. In the first section of this chapter, we introduce intrinsic metrics in the context of graphs and discuss basic properties and examples. Secondly, we study Liouville theorems with respect to l p bounds in the spirit of Yau and Karp. These theorems are used to determine the domain of the generators on l p. Furthermore, we prove a result on essential selfadjointness in the spirit of Gaffney. Then we turn to spectral estimates. First, a lower bound on the bottom of the spectrum is presented by means of an isoperimetric constant. This Cheeger inequality solves a problem addressed by Dodziuk/Karp in Upper bounds by exponential volume growth rates in the sense of Brooks and 35

38 36 2. INTRINSIC METRICS Sturm are discussed afterwards. Finally, we look into the question of p-independence of the generators on l p. Such investigations have their origin in a question of Simon for Schrödinger operators and are found for manifolds in the work of Sturm. Substantial parts of the presentation of this chapter are taken from the survey article [Kel14b] Definition and basic facts In this section, we introduce the concept of intrinsic metrics for graphs. We compare this concept to other metrics that appear in the literature. Furthermore, we present a Hopf Rinow theorem for path metrics on locally finite graphs and discuss important conditions which provide a suitable framework in the general case Definition. We call a symmetric map ρ : X X [0, ) with zero diagonal a pseudo metric if it satisfies the triangle inequality. In [FLW14, Definition 4.1] a definition of intrinsic metrics is given for general regular Dirichlet forms. It can be seen by [FLW14, Lemma 4.7, Theorem 7.3] that the definition below coincides with that in [FLW14]. Definition 2.1. A pseudo metric ρ is called an intrinsic metric with respect to a graph (b, c) over (X, m) if b(x, y)ρ 2 (x, y) m(x), x X. y X Notions similar to such metrics were introduced in the context of graphs or jump processes under the name adapted metrics in [Fol14a, Fol14b, GHM12, Hua11a, HS14, MU11] Examples and relations to other metrics. In this subsection, we explore the definition of intrinsic metrics through examples and counter examples The degree path metric. A specific example of an intrinsic metric was introduced by Huang [Hua11a, Definition 1.6.4] and also appeared in [Fol11]. Let the pseudo metric ρ 0 : X X [0, ) be given by ρ 0 (x, y) = inf x=x 0 x 1... x n=y n i=1 ( Deg(x i 1 ) Deg(x i ) ) 1 2, x X, where Deg : X (0, ) is the weighted vertex degree defined as Deg(x) = 1 b(x, y), x X. m(x) y X We call any metric that minimizes sums of weights over paths of edges a path metric.

39 2.1. DEFINITION AND BASIC FACTS 37 It can be seen directly that ρ 0 is an intrinsic metric for the graph (b, c) over (X, m): b(x, y)ρ 2 0(x, y) b(x, y) Deg(x) Deg(y) b(x, y) Deg(x) = m(x). y X y X y X There is the following intuition behind the definition of ρ 0. Consider the Markov process (X t ) t 0 associated to the semigroup e tl via e tl f(x) = E x (f(x t )), x X, where E x is the expected value conditioned on the process starting at x. The random walker modeled by this process jumps from a vertex x to a neighbor y with probability b(x, y)/ z b(x, z). Moreover, the probability of not having left x at time t is given by P x (X s = x, 0 s t) = e Deg(x)t. Qualitatively, this indicates that the larger Deg(x), the faster the random walker leaves x. Looking at the definition of ρ 0 (x, y), the larger the degree of either x or y the closer they are. Combining these two observations, we see that the faster the random walker jumps along an edge, the shorter the edge is with respect to ρ 0. Of course, the jumping time along an edge connecting x to y is not symmetric and depends on whether one jumps from x to y or from y to x as the degrees of x and y can be very different. In order to get a symmetric function, ρ 0 favors the vertex with the larger degree and the faster jumping time. There is an analogy to the Riemannian setting in terms of mean exit times of small balls. Consider a small ball B r of radius r on a d-dimensional Riemannian manifold, then the first order term of the mean exit time of B r is r 2 /2d [Pin85]. On a locally finite graph for a vertex x a small ball with respect to ρ 0 can be thought to have radius r = inf y x ρ 0 (x, y)/2, namely this ball contains only the vertex itself. Now, computing the mean exit time of this ball gives 1/Deg(x) r 2, where equality holds whenever Deg(x) = max y x Deg(y) The combinatorial graph distance. Next, we come to a metric that is often the most immediate choice when one considers metrics on graphs. That is the combinatorial graph distance which is the path metric defined as d(x, y) = min #{k N 0 there exist x 0,..., x k with x = x 0... x k = y}. The next lemma shows that the combinatorial graph distance is equivalent to an intrinsic metric if and only if the graph has bounded geometry. This fact was already observed in [FLW14, KLSW]. Lemma 2.2. Let (b, c) be a graph over (X, m). The following are equivalent:

40 38 2. INTRINSIC METRICS (i) The combinatorial graph distance d is equivalent to an intrinsic metric. (ii) Deg is a bounded function. Furthermore, if additionally c 0, then also the following is equivalent: (iii) L is a bounded operator. Proof. (i) (ii): Let ρ be an intrinsic metric such that C 1 ρ d Cρ. Then, b(x, y) = b(x, y)d 2 (x, y) C 2 b(x, y)ρ 2 (x, y) C 2 m(x), x X x X x X for all x X. Hence, Deg C 2. (ii) (i): Assume Deg C and consider the degree path metric ρ 0. Then, ρ 1 = ρ 0 1 is an intrinsic metric as well. Clearly, ρ 1 d. On the other hand, by Deg C we immediately get ρ 1 C 1 2 d. The equivalence (ii) (iii) follows from Theorem 1.5. The theorem implies, in particular, that, in the case of the normalizing measure m = n, the combinatorial graph distance is an intrinsic metric as Deg = n/m = 1, in the case of c 0, and Deg n/m = 1 in general. Furthermore, for a graph with standard weights and the counting measure associated to the Laplacian, the combinatorial graph distance d is a multiple of an intrinsic metric if and only if the combinatorial vertex degree deg is bounded since Deg = deg Comparison to the strongly local case. An important difference to the case of strongly local Dirichlet forms is that, in the graph case, there is no maximal intrinsic metric. For example, for a complete Riemannian manifold M the Riemannian distance d M is the maximal C 1 metric ρ M that satisfies M ρ M (o, ) 1, for all o M, where M is the Riemannian gradient. In fact, d M can be recovered by the formula d M (x, y) = sup{f(x) f(y) f C c (M), M f 1}, x, y X. Now, for discrete spaces, the maximum of two intrinsic metrics is not necessarily an intrinsic metric. This is discussed next. Consider the pseudo metric σ where σ(x, y) = sup{f(x) f(y) f A}, x, y X, A = { f : X R y X b(x, y) f(x) f(y) 2 m(x) for all x X }.

41 2.1. DEFINITION AND BASIC FACTS 39 As discussed for the Riemannian case above, in the strongly local case the analogue of σ defines the maximal intrinsic metric. But σ is, in general, not even equivalent to an intrinsic metric in the graph case. A basic example where this can be immediately seen can be found in [FLW14, Example 6.2]. More generally, this phenomena can be checked for arbitrary tree graphs with standard weights and counting measure to which the the operator is associated. In this case, σ = 1d 2 and, by the discussion above, we already know that the combinatorial graph distance d is not equivalent to an intrinsic metric whenever is unbounded. An abstract way to see that σ is, in general, not intrinsic is discussed in [KLSW, Section 1]. Namely, the set of Lipschitz continuous functions Lip ρ with respect to an intrinsic metric ρ is included in A and Lip ρ is closed under taking suprema. On the other hand, A is, in general, not closed under taking suprema. Hence, in general, Lip ρ is not equal to A. It would be interesting to know whether one can characterize the situation when these two spaces are different Another path metric. Colin de Verdiere/Torki-Hamza/Truc [CdVTHT11a] studied a path pseudo metric δ which is given as δ(x, y) = n 1 inf x=x 0... x n=y i=0 ( m(x) m(y) b(x, y) ) 1 2, x, y X. By a similar argument as in Lemma 2.2, this metric can be seen to be equivalent to the intrinsic metric ρ 0 if and only if the combinatorial vertex degree deg is bounded on the graph A Hopf-Rinow theorem. We shall stress that, in general, an intrinsic metric ρ (and in particular ρ 0 ) is not a metric and (X, ρ) might not even be locally compact. This can be seen from examples in [HKMW13, Example A.5]. However, for locally finite graphs and path metrics such as ρ 0, the situation is much more tame. For example the topology is discrete in this case. In [HKMW13] a Hopf-Rinow type theorem is shown, see also [Mil11]. Recall that for a path γ = (x n ) the length with respect to a metric ρ is given by l(γ) = j 0 ρ(x j, x j+1 ) and, moreover, a path γ = (x n ) is called a geodesic with respect to a metric ρ if ρ(x 0, x n ) = l((x 0,..., x n )) for all n. Theorem 2.3 (Theorem A1 in [HKMW13]). Let (b, c) be a locally finite graph over (X, m) and let ρ be a path metric. Then, the following are equivalent: (i) (X, ρ) is complete as a metric space. (ii) (X, ρ) is geodesically complete, that is any infinite geodesic has infinite length. (iii) The balls in (X, ρ) are finite.

42 40 2. INTRINSIC METRICS Some important conditions. As discussed above, the topology induced by an intrinsic metric can be rather wild. So, in the general situation, one often has to make additional assumptions. Here, we present some of the most important assumptions and discuss their implications. We say a pseudo metric ρ admits finite balls if (iii) in the theorem above is satisfied for ρ, i.e., if (B) The distance balls B r (x) = {y X ρ(x, y) r} are finite for all x X, r 0. A somewhat weaker assumption is that the weighted vertex degree is bounded on balls: (D) The restriction of Deg to B r (x) is bounded for all x X, r 0. Clearly, (B) implies (D). Moreover, (D) is equivalent to the fact that L restricted to the l 2 space of a distance ball is a bounded operator, confer Theorem 1.5. The assumptions (B) and (D) can be understood as bounding ρ in a certain sense from below. Next, we come to an assumption which may be understood as an upper bound. We say a pseudo metric ρ has finite jump size if (J) The jump size s = sup{ρ(x, y) x, y X, x y} is finite. The assumptions (B) and (J) combined have the following consequence on the combinatoric structure of the graph. Lemma 2.4. Let (b, c) be a graph over (X, m) and let ρ be an pseudo metric. If ρ satisfies (B) and (J), then the graph is locally finite. Proof. If there was a vertex with infinitely many neighbors, then there would be a distance ball containing all of them by finite jump size. However, this is impossible by (B) Liouville theorems The classical Liouville theorem in R n states that if a harmonic function is bounded from above, then the function is constant. Here, we look into boundedness assumptions such as l p growth bounds and present results proven in [HK14]. In Section we have already presented such a Liouville theorem under the assumption of uniformly positive measure. Here, we address the question of arbitrary measures under some completeness assumption on the graph. Such results go back to Yau [Yau76] and Karp [Kar82] in the case of manifolds. We first discuss the results for manifolds in the next subsection. Afterwards, we discuss the case of the normalized Laplacian for graphs and the results that have been proven for this operator. Finally, we present theorems of [HK14] that recover Yau s and Karp s results for

43 2.2. LIOUVILLE THEOREMS 41 general graph Laplacians using intrinsic metrics. As a consequence, this yields a sufficient criterion for recurrence. The results of this section are used later to address questions such as essential selfadjointness and to determine the domain of the generators. Throughout this section, keep in mind that absence of non-constant positive subharmonic functions implies the absence of non-constant harmonic functions, see Lemma Historical remarks. Here, we discuss the results that preceded [HK14] in the case of manifolds and graphs Manifolds. Let M be a connected Riemannian manifold and M the Laplace Beltrami operator. A twice continuously differentiable function f on M is called harmonic (respectively, subharmonic) if M f = 0 (respectively, M f 0). The assumption that f is twice continuously differentiable can be relaxed, but here we do not want to put the focus on the degree of smoothness. In 1976 Yau [Yau76] proved that on a complete Riemannian manifold M any harmonic function or positive subharmonic function in L p (M) is constant. This result was later strengthened by Karp in 1982 [Kar82]. Namely, any harmonic function or positive subharmonic function f that satisfies r inf dr =, r 0 >0 r 0 f1 Br p p is already constant, where 1 Br is the characteristic function of the geodesic ball B r about some arbitrary point in the manifold. Yau s theorem is a direct consequence of Karp s result. Later in 1994 Sturm [Stu94] generalized Karp s theorem to the setting of strongly local Dirichlet forms, where balls are taken with respect to the intrinsic metric. The underlying assumption on the metric is that it generates the original topology and all balls are relatively compact Graphs. For graphs b over (X, m), so far results in this direction were obtained for the normalizing measure m = n only. In this case, the operator L is bounded, see Section 1.5, and the combinatorial graph distance d is an intrinsic metric, see Subsection (Of course, harmonicity depends only on the graph b and not on the measure m, but for the function to be in an l p space does depend on the measure.) Starting 1997 with Holopainen/Soardi [HS97], Rigoli/Salvatori/ Vignati [RSV97], Masamune [Mas09], eventually in 2013 Hua/Jost [HJ13] showed that if a harmonic or positive subharmonic function f satisfies 1 lim inf r r f1 2 B r(x) p p <,

44 42 2. INTRINSIC METRICS for some p (1, ) and x X, then f must be constant. Here, the balls are taken with respect to the combinatorial graph distance. This directly implies Yau s theorem for p (1, ). Moreover, Hua/Jost [HJ13] also show Yau s theorem for p = Yau s and Karp s theorem for general graphs. We now turn to general graphs equipped with an intrinsic metric. As in the manifold setting, we need a completeness assumption on the graph as a metric space. For graphs with the normalizing measure, completeness is guaranteed since the combinatorial graph distance always gives rise to a complete metric space. In the theorem below, we state a graph version of Yau s theorem for the general graphs. We assume that the weighted vertex degree is bounded on balls (D) and the jump size is finite (J). In the case of a path metric on a local finite graph, the Hopf- Rinow theorem, Theorem 2.3, shows that metric completeness implies (D). Theorem 2.5 (Corollary 1.2 in [HK14]). Let b be a graph over (X, m) and let ρ be an intrinsic metric with bounded degree on balls (D) and finite jump size (J). If f l p (X, m), p (1, ), is a positive subharmonic function then f is constant. Let us mention that the case l 1 is more subtle in the general case. In [HK14, Theorem 1.7] it was shown that for stochastically complete graphs Yau s Liouville theorem remains true in the case p = 1. The proof follows ideas from [Gri99]. Otherwise, there are counter examples, see [HK14, Section 4]. Next, we turn to a graph version of Karp s theorem which is the main result of [HK14]. Theorem 2.6 (Theorem 1.1 in [HK14]). Let b be a graph over (X, m) and let ρ be an intrinsic metric with bounded degree on balls (D) and finite jump size (J). If f is a positive subharmonic function such that for some p (1, ) and x X inf r 0 >0 r 0 r f1 Br(x) p p dr =, then f is constant. Here, 1 B is again the characteristic function of a set B X. In particular, the theorem above implies the result of Hua/Jost [HJ13]. It can even be seen that a harmonic function f satisfying lim sup r 1 r 2 log r f1 B r(x) p p <, for some p (1, ), is constant. The proof of the above theorems uses a Caccioppoli inequality, [HK14, Theorem 1.8]. This itself is proven by the use of suitable

45 2.3. DOMAIN OF THE GENERATORS 43 cut-off functions stemming from the intrinsic metrics. The part of the chain rule is then played by the inequality f p 1 (x) f p 1 (y) C(f(x) f(y)) p 1 (f(x) f(y)), x, y X which is taken from [HS14]. The proof of Theorem 2.6 then uses an induction argument such as [Stu94] Recurrence. As a direct consequence of Karp s theorem we get a sufficient criterion for recurrence of a graph. A connected graph b over X is called recurrent if for all measures m and some (all) x, y X, we have 0 e tl 1 {x} (y)dt =. This is equivalent to absence of non-constant bounded subharmonic functions. Analogous results to the criterion below in the case of manifolds and strongly local Dirichlet forms are due to [Kar82, Theorem 3.5] and [Stu94, Theorem 3]. For graphs the result below generalizes the results of [DK86, Theorem 2.2], [RSV97, Corollary B], [Woe00, Lemma 3.12], [Gri09, Corollary 1.4], [MUW12, Theorem 1.2]. Theorem 2.7 (Corollary 1.6 in [HK14]). Let b be a connected graph over (X, m) and let ρ be an intrinsic metric with bounded degree on balls (D) and finite jump size (J). If for some x X then the graph is recurrent. 1 r dr =, m(b r (x)) 2.3. Domain of the generators and essential selfadjointness In this section we address the question of identifying the domain of the generators L p. Classically, the special case p = 2 received particular attention. Going back to investigations of Friedrichs and von Neumann, a classical question is whether a symmetric operator on a Hilbert space has a unique selfadjoint extension. This property is often studied under the name essential selfadjointness. The connection of essential selfadjointness with metric completeness is that if there exists a boundary, one might have to impose certain boundary conditions in order to obtain a selfadjoint operator. We first discuss the manifold case which is often referred to as Gaffney s theorem. Secondly, we consider graphs and recover Gaffney s theorem with the use of intrinsic metrics. Furthermore, we determine the domain of the generators L p on l p. The results of this section are found in [HKMW13] and [HK14].

46 44 2. INTRINSIC METRICS Historical remarks. Again, we discuss some of the results that preceded [HKMW13] and [HK14] in the case of manifolds and graphs Manifolds. A result going back to the work of Gaffney [Gaf51, Gaf54] states that, on a geodesically complete manifold, the so-called Gaffney Laplacian is essentially selfadjoint. This is equivalent to the uniqueness of the Markovian extension of the minimal Laplacian. Independently, essential selfadjointness of the Laplace Beltrami operator on the compactly supported, infinitely often differentiable functions was shown by Roelcke [Roe60]. For later results in this direction see also [Che73, Str83] Graphs. The first results connecting metric completeness and essential selfadjointness were obtained by Torki-Hamza [TH10], Colin de Verdière/ Torki-Hamza/Truc [CdVTHT11a, CdVTHT11b] and Milatovic [Mil11, Mil12]. These results were proven for (magnetic) Schrödinger operators on graphs with bounded combinatorial vertex degree and the metric δ discussed in Section As discussed there, δ is equivalent to an intrinsic metric if and only if the combinatorial vertex degree is bounded Gaffney s theorem for graphs. For the general case of unbounded combinatorial vertex degree, we consider intrinsic metrics to recover a Gaffney theorem. In Lemma 1.6 we demonstrated that we may not have LC c (X) l 2 (X, m) for general graphs. Hence, in general, L is not a symmetric operator on C c (X) as a subspace of l 2 (X, m). Nevertheless, one can still determine whether the forms with Dirichlet and Neumann boundary conditions are equal. Recall that we refer to the restriction of Q to the closure of C c (X) as the form with Dirichlet boundary conditions Q = Q (D) and to the restriction of Q to D l 2 (X, m) as the form with Neumann boundary conditions Q (N). Moreover, in the case of equality, we can even identify the domain of the generator. The following result is found in [HKMW13] for graph Laplacians and in [GKS12] for magnetic Schrödinger operators. Theorem 2.8 (Theorem 1 in [HKMW13]). Let b be a graph over (X, m) and let ρ be an intrinsic metric with bounded degree on balls (D) and finite jump size (J). Then, and Q (D) = Q (N) D(L) = {f l 2 (X, m) Lf l 2 (X, m)}. Furthermore, if LC c (X) l 2 (X, m), then L Cc(X) is essentially selfadjoint on l 2 (X, m).

47 2.4. ISOPERIMETRIC CONSTANTS AND LOWER SPECTRAL BOUNDS 45 Here, the assumptions (D) and (J) serve again as an analogue for the completeness assumption. By the virtue of the Hopf Rinow type theorem, Theorem 2.3, we immediately get the following analogue to the classical Gaffney theorem from Riemannian geometry. Corollary 2.9 (Theorem 2 in [HKMW13]). Let b be a locally finite graph over (X, m) and let ρ be an intrinsic path metric. If (X, ρ) is metrically complete, then L Cc(X) is essentially selfadjoint on l 2 (X, m). We can also determine the domain of the generators of the semigroup on l p. Theorem 2.10 (Corollary 1.4 in [HK14]). Let b be a graph over (X, m) and let ρ be an intrinsic metric with bounded degree on balls (D) and finite jump size (J). Then, D(L p ) = {f l p (X, m) Lf l p (X, m)}, for all p (1, ). Furthermore, in [HKMW13], the case of metrically incomplete graphs is treated. For locally finite graphs the capacity of the Cauchy boundary is defined. Whenever the boundary has finite capacity equality of the form with Dirichlet and Neumann boundary conditions can be characterized by the boundary having zero capacity [HKMW13, Theorem 3]. It is also shown that if the upper Minkowski codimension of the boundary is larger than 2, then the boundary has zero capacity [HKMW13, Theorem 4] Isoperimetric constants and lower spectral bounds We now turn to the spectral theory of the operator L. In this section we aim for lower bounds on the bottom of the spectrum λ 0 (L) = inf σ(l) via so called isoperimetric estimates. Such lower bounds are often referred to as Cheeger s inequality. We first discuss the result on manifolds going back to Cheeger from Then, we discuss how an analogous result was proven in the 80 s for the normalized Laplacian by Dodziuk/Kendall and what kind of problems occur for the operator. Finally, we examine how intrinsic metrics can be used to overcome these problems and establish this inequality for general graph Laplacians which is proven in [BKW14] Historical remarks on Cheeger s inequality Manifolds. For a non-compact Riemannian manifold M the isoperimetric constant or Cheeger constant is defined as h M = inf S Area( S) vol(int(s)),

48 46 2. INTRINSIC METRICS where S runs over all hypersurfaces cutting M into a precompact piece int(s) and an unbounded piece. Denote by λ 0 ( M ) the bottom of the spectrum of the Laplace-Beltrami operator. The well known Cheeger inequality reads as λ 0 ( M ) h2 M 4. See [Che70] for Cheeger s original work on the compact case and [Bro93] for a discussion of the non-compact case Graphs with standard weights. There is an enormous amount of literature on isoperimetric inequalities, especially for finite graphs, see [AM85] as one of the first papers for finite graphs. Here, we restrict ourselves to infinite graphs although the methods can also be applied to the finite graph case as well. Recall that the boundary of a set W X is defined as the set of edges emanating from W, i.e., W = {(x, y) W X \ W x y}. In 1984 Dodziuk [Dod84] considered graphs with standard weights and the counting measure. The isoperimetric constant he studied is closely related to W h 1 = inf W X finite W and Dodziuk s proof yields λ 0 ( ) h2 1 2D, with D = sup x X deg(x). This analogue of Cheeger s inequality is effective for graphs with bounded vertex degree. However, for unbounded vertex degree, the bound becomes trivial. Two years later Dodziuk and Kendall [DK86] proposed a solution to this issue by considering graphs with standard weights and the normalizing measure n = deg instead. The corresponding isoperimetric constant is h n = inf W X finite W deg(w ) and they prove in [DK86] for the normalized Laplacian n that λ 0 ( n ) h2 n 2. This analogue of Cheeger s inequality does not have the disadvantage of becoming trivial for unbounded vertex degree. This seems to be the reason that in the following the operator was rather neglected in spectral geometry of graphs and the normalized Laplacian n gained momentum.

49 2.4. ISOPERIMETRIC CONSTANTS AND LOWER SPECTRAL BOUNDS Cheeger s inequality for graphs. The considerations in the previous sections suggest that intrinsic metrics allow results for general graph Laplacians. However, it is not obvious how isoperimetric constants can be related to a specific metric. So, the crucial new element is to see how a metric is already hidden in the previous definition of isoperimetric constants which worked for the normalized Laplacian. Revisiting the definition of the area of the boundary above, we find that b( W ) = b(x, y) = b(x, y)d(x, y) (x,y) W (x,y) W with the combinatorial graph distance d on the right hand side. Remember that d is an intrinsic metric for the graph b over (X, n). The new idea is to replace d by an intrinsic metric ρ for a graph b over (X, m). We define Area( W ) = b(x, y)ρ(x, y). (x,y) W That is, we take the length of an edge into consideration to measure the area of the boundary. We define h = inf W X finite Area( W ), m(w ) and obtain the following theorem which is found in [BKW14]. Theorem 2.11 (Theorem 1 in [BKW14]). Let b be a graph over (X, m) and let ρ be an intrinsic metric. Then, λ 0 (L) h2 2. The interesting new part of the theorem is the definition of the isoperimetric constant. Having this definition the usual proof scheme applies which is sketched below. Idea of the proof. The proof of the theorem is based on an area and a co-area formula. For f 0, let Ω t = {x X f(x) > t}. Then, one can prove, using Fubini s theorem for f C c (X), m(ω t ) = x X f(x)m(x), Area( Ω t ) = x,y X b(x, y)ρ(x, y) f(x) f(y). The rest of the proof is basically the Cauchy-Schwarz inequality and various algebraic manipulations.

50 48 2. INTRINSIC METRICS One may also consider potentials c 0 in the estimate by introducing edges from vertices x with c(x) > 0 to virtual sibling vertices ẋ with edge weight b(x, ẋ) = c(x). The union of vertices x X and ẋ is denoted by Ẋ. Furthermore, we extend an intrinsic metric ρ on X to the new edges via ρ(x, ẋ) = (m(x) y X b(x, y)ρ(x, y)2 ) 1 2 c(x) The extension of ρ becomes an intrinsic metric if one chooses m(ẋ) = m(x). Now, we define h by taking the infimum of the quotient with the extension of b and ρ as above only over subsets of X, see [BKW14, Section 5] Volume growth and upper spectral bounds In this section, we discuss upper bounds for the bottom of the essential spectrum λ ess 0 (L) = inf σ ess (L). The essential spectrum σ ess (L) of an operator is the part of the spectrum which does not contain discrete eigenvalues of finite multiplicity. Clearly, λ 0 (L) λ ess 0 (L). We first discuss the classical result on Riemannian manifolds going back to Brooks and Sturm. There are corresponding results for the normalized Laplacian. Next, we show how such a result fails in the case of the Laplacian with respect to the counting measure based on examples developed in [KLW13]. Finally, we employ intrinsic metrics to recover Brooks result for general graph Laplacians based on results of [HKW13]. Let us remark that the results in [HKW13] are proven in the general context of regular Dirichlet forms Historical remarks Manifolds. Let M be a complete connected non-compact Riemannian manifold with infinite volume. Let λ ess 0 ( M ) be the bottom of the essential spectrum of the Laplace Beltrami operator M. Let µ M be the upper exponential growth rate of the distance balls 1 µ M = lim sup r r log vol(b r(x)), for an arbitrary x M. Brooks showed in 1981 [Bro81] that λ ess 0 ( M ) µ2 M 4. Later, in 1996, Sturm [Stu94] showed using the lower exponential growth rate of the distance balls with variable center µ M = lim inf r inf x M 1 r log vol(b r(x)).

51 2.5. VOLUME GROWTH AND UPPER SPECTRAL BOUNDS 49 the following bound λ 0 ( M ) µ2 M 4. Indeed, the result in [Stu94] is shown in the general context of strongly local regular Dirichlet forms. An immediate corollary is that for M with subexponential growth, i.e., µ M = 0, the value 0 is in the spectrum of the Laplace Beltrami operator Graphs. For graphs with standard weights and the normalizing measure Dodziuk/Karp [DK88] proved in 1987 the first analogue of Brooks theorem for graphs. This result was later improved by Ohno/Urakawa [OU94] and Fujiwara [Fuj96a] resulting in the estimate with λ ess 0 ( n ) 1 2eµn/2 e µn + 1 µ n = lim sup r 1 r log n(b r(x)), for arbitrary x X and n = deg. It can be checked that the bound above is smaller than µ 2 n/8. Next, we discuss how, for graphs with standard weights and the counting measure, such a bound fails when volume growth is considered via the combinatorial graph distance. The examples are so called anti-trees which were studied by Wojciechowski [Woj09] as counter examples for volume bounds for stochastic completeness. Specifically, anti-trees are highly connected graphs. They can be characterized as follows: A vertex in a sphere (with respect to a root vertex) is connected to every neighbor in the succeeding sphere, (where spheres are considered with respect to the combinatorial graph distance). See Figure 1 below for an example. Figure 1. An anti-tree with s r+1 = 2 r

52 50 2. INTRINSIC METRICS For an anti-tree, let s r be the number of vertices with combinatorial graph distance r to a root vertex. Denote, furthermore, v r = s s r, r 0. In [KLW13] (confer Theorem 1.18) it was shown that ( v ) 1 r a = s r s r+1 r=0 is a lower bound on the bottom of the spectrum λ 0 ( ) of (where a = 0 if the sum diverges). Moreover, in the case where the sum converges, the spectrum of is purely discrete, i.e., there is no essential spectrum. In particular, this result implies that anti-trees with s r r 2+ε, ε > 0, have positive bottom of the spectrum and purely discrete spectrum, see [KLW13, Section 6]. However, for s r r 2+ε, we have v r r 3+ε, that is, these are graphs of little more than cubic growth with positive bottom of the spectrum and no essential spectrum. Hence, there is no analogue to Brooks or Sturm s theorem for with respect to the combinatorial graph distance Brooks theorem for graphs. Let b be a graph over (X, m) and let ρ be an intrinsic metric. Let B r (x) be the distance r ball about a vertex x with respect to the metric ρ. We define for fixed x X and µ = lim inf r µ = lim inf r 1 r log m(b r(x)), inf 1 x X r log m(b r(x)). In [HKW13] analogues of Brooks and Sturm s theorem are proven for regular Dirichlet forms. As a special case the following theorem is obtained for graphs. Folz [Fol14b] independently proved, by different methods, a special case of the theorem below for locally finite graphs with uniformly positive measure. Theorem 2.12 (Corollary 4.2 in [HKW13]). Let b be a connected graph over (X, m) and let ρ be an intrinsic metric such that the balls are finite (B). Then, λ 0 (L) µ2 8. If furthermore m(x) =, then λ ess 0 (L) µ2 8. The idea of the proof combines ideas of [Stu94] and a Perrson-type theorem.

53 2.6. VOLUME GROWTH AND l p -INDEPENDENCE OF THE SPECTRUM 51 1 Idea of the proof. Let µ = lim sup log m(b r r r(x)). Then, the functions f a = e aρ(o, ) for a > µ/2 and fixed o X are in l 2 (X, m). Moreover, by the mean value theorem and the intrinsic metric property we find that Q(f a ) a2 2 f a (x) 2 y X x X To pass from µ to µ or µ we consider g a,r = (e 2ar f a 1) 0. b(x, y)ρ(x, y) 2 a2 2 f a 2 Note that g a,r is supported on B 2r and, therefore, g a,r is in C c (X) whenever (B) applies. Finally, to see the statement for the essential spectrum, we need to modify g a,r such that we obtain a sequence of functions that converge weakly to zero. We achieve this by cutting off g a,r at 1 on B r, i.e., h a,r = 1 g a,r. The weak convergence of h a,r to zero is ensured by the assumption m(x) =. Now, the statement follows by a Persson-type theorem [HKW13, Proposition 2.1]. We end this section with a few remarks. Remark. (a) In [HKW13] it is also shown that the assumption (B) can be replaced by the assumption (M*) that any infinite path has infinite measure from Section 1.3. (b) As a corollary, we get under the assumptions of the theorem 2h µ for the Cheeger constant h defined in Section (c) By comparing the degree path metric ρ 0 with the combinatorial graph distance d on anti-trees one finds that for s r r 2 ε the balls with respect to ρ 0 grow polynomially, for s r r 2 they grow exponentially and for s r r 2+ε the graph has finite diameter with respect to ρ 0. This shows that the examples in the section above are indeed sharp Volume growth and l p -independence of the spectrum In this section we turn to the spectra of the operators L p on l p, p [1, ] which were defined in Section In the beginning of the 80 s Simon [Sim82] asked the famous question whether the spectra of certain Schrödinger operators on R d are independent on which L p space they are considered. Hempel/Voigt [HV86, HV87] gave an affirmative answer in Here, we consider a geometric analogue of this question. This goes back to a theorem of Sturm on Riemannian manifolds, [Stu94]. Here, we deal with graphs. With an intrinsic metric at hand, a corresponding result was obtained in [BHK13] which is discussed afterwards.

54 52 2. INTRINSIC METRICS Historical remarks. In 1993, Sturm [Stu94] proved a theorem for uniformly elliptic operators on a complete Riemannian manifold M whose Ricci curvature is bounded below. We assume that M has uniform subexponential growth, i.e., for any ε > 0 there is C > 0 such that for all r > 0 and all x M vol(b r (x)) Ce εr vol(b 1 (x)). Then the spectrum of a uniformly elliptic operator on such a manifold is independent of the space L p (M), p [1, ], on which it is considered Sturm s theorem for graphs. A graph (b, c) over (X, m) with an intrinsic metric ρ is said to have uniform subexponential growth if for any ε > 0 there is C > 0 such that for all r > 0 and all x M m(b r (x)) Ce εr m(x). The proof of the following theorem follows the strategy of Sturm in [Stu94]. Theorem 2.13 (Theorem 1 in [BHK13]). Let (b, c) be a connected graph over (X, m) and let ρ be an intrinsic metric such that the balls are finite (B), which has finite jump size (J) and the graph has uniform subexponential growth. Then, σ(l p ) = σ(l 2 ), p [1, ]. Remark. (a) A question in the direction of p-independence of the spectrum for graphs was already brought up by Davies [Dav07, p. 378]. (b) In contrast to Sturm s result for manifolds, no curvature type assumption is needed in the theorem above. Indeed, there are graphs with unbounded weighted vertex degree which satisfy the assumptions, see [BHK13, Example 3.2]. On the other hand, the assumptions of the theorem already imply that the combinatorial vertex degree must be bounded, see [BHK13, Lemma 3.1]. (c) The statement of the theorem is, in general, wrong if one drops the growth assumption. This was already discussed in Section On the other hand, it is an open question what happens for graphs that are subexponentially growing, i.e., µ = 0, but not uniformly subexponentially growing.

55 CHAPTER 3 Curvature on planar tessellations In this chapter, we survey results relating curvature bounds, geometry and spectral theory that are proven in the original manuscripts [Kel10, Kel11, KP11, BGK13, BHK13]. Our focus lies on infinite planar tessellations which can be considered as discrete analogues of non compact surfaces. The tiles of the tessellations shall be seen as regular polygons. We study a curvature function that arises as an angular defect and satisfies a Gauß Bonnet formula. This idea goes back at least to Descartes, see [Fed82], and appeared since then independently at various places, see e.g. [Sto76, Gro87, Ish90, Woe98]. A systematic study of geometric properties of tessellations with non-positive curvature was undertaken by Baues/Peyerimhoff [BP01, BP06]. Furthermore, a substantial amount of research was conducted on various topics for tessellations in dependence of the curvature, see e.g. [Blo10, Che09, CC08, DM07, Hig01, HJ, HJL, Kel10, KP11, Oh13, Sto76, SY04, Woe98, Żuk97]. The operators of interest are graph Laplacians with standard weights. First, we show spectral bounds resulting from curvature bounds. Here, the quantitative bounds result from estimates on an isoperimetric constant and a volume growth rate, see [KP11]. Secondly, we take a closer look at the case of uniformly unbounded negative curvature. This is equivalent to discreteness of spectrum, [Kel10], and we present eigenvalue asymptotics [BGK13] in this case. Thirdly, we summarize results on the p-independence of the spectrum of the Laplacian as an operator on l p, p [1, ], from [BHK13]. Parts of the exposition of this chapter are taken from the survey article [Kel14b]. One can also define a related notion of curvature for general planar graphs. By the virtue of [Kel11] one sees that non-positive curvature implies that the graph is almost a tessellation (possibly with unbounded tiles intersecting in a path of edges). With these considerations most results for tessellations can be extended to general planar graphs. As this approach is more technical and at some points less geometrically intuitive, we only discuss it at the end. 53

56 54 3. CURVATURE ON PLANAR TESSELLATIONS 3.1. Set up and definitions In this section we introduce planar tessellations, notions of curvature and recall the definition of the graph Laplacian Planar tessellations. In this chapter we consider graphs with standard weights. So, we adapt our notation of the previous chapters to the notation that is classically used in this context. The vertex set X is still a countable discrete set. Let b be a graph with standard weights over X. That is, b takes values in {0, 1} and the function c vanishes. We introduce the set of edges as subsets of X with two elements as follows E = {{x, y} X b(x, y) = 1}. A graph is called planar if there is an orientable topological surface S that is homeomorphic to R 2 such that the graph can be embedded without self intersections into S. The vertices X are mapped to points in S and the edges E to line segments in S connecting vertices. In the following we will identify a combinatorial planar graph with its embedding and denote it by (X, E). Nevertheless, we stress that we only use the combinatorial properties of the graph which do not depend on the embedding. A graph is locally compact if there is an embedding into S such that for every compact K S, one has #{e E e K } <. Next, we introduce the set of faces F that has the connected components of S \ E as elements. For f F, we denote by f the closure of f in S. We write G = (X, E, F ) for locally compact planar graphs and, following [BP01, BP06], we call G = (X, E, F ) a tessellation if the following three assumptions are satisfied: (T1) Every edge is contained in two faces. (T2) Two faces are either disjoint or intersect in a vertex or an edge. (T3) Every face is homeomorphic to the unit disc. There are related definitions such as semi-planar graphs see [HJ, HJL] and locally tessellating graphs [Kel11]. Indeed, most of the results presented here hold for general planar graphs on surfaces of finite genus. However, the definition of curvature becomes more involved and some of the estimates turn out to be more technical. We refer to Section 3.5 for corresponding considerations for planar graphs.

57 3.1. SET UP AND DEFINITIONS Curvature. Let G = (X, E, F ) be a tessellation. In order to define a curvature function, we first introduce the vertex degree and the face degree. We denote the vertex degree of a vertex v X by v = deg(v) = #edges emanating from v. We use the notation v if we use vertex degree geometrically and deg(v) if we use it analytically. The face degree of a face f F is defined as f = #boundary edges of f = #boundary vertices of f. The vertex curvature κ : X R is defined as κ(v) = 1 v f. f F,v f The idea traces back to Descartes [Fed82] and was later introduced in the above form by Stone in [Sto76] who refers to ideas of Alexandrov. Since then, this notion of curvature reappeared at various places, e.g. [Gro87, Ish90] and was widely used, see e.g. [BP01, BP06, DM07, Hig01, HJL, Kel10, Kel11, KP11, Oh13, Woe98, Żuk97]. The notion of curvature is motivated by an angular defect: Assume a face f is a regular polygon. Then, the inner angles of f are all equal to β(f) = 2π f 2. 2 f This formula is easily derived: Walking around f once results in an angle of 2π, while going around the f corners of f one takes a turn by an angle of π β(f) each time. In this light, the vertex curvature may be rewritten as 2πκ(v) = 2π β(f), v X. f F,v f It shall be stressed that the mathematical nature of κ is purely combinatorial. Nevertheless, thinking of the tessellation with a suitable embedding allows for a geometric interpretation. The notion has its further justification in the Gauß-Bonnet formula relating the sum of the curvatures of a simply connected set to the Euler characteristic. This formula is mathematical folklore and may, for instance, be found in [BP01] or [Kel11]. We next consider a finer notion of curvature. Asking which contribution to the total curvature at a vertex v comes from the corner at a face f with v f gives rise to the corner curvature. Precisely, the set of corners of a tessellation G is given by C(G) = {(v, f) X F v f}.

58 56 3. CURVATURE ON PLANAR TESSELLATIONS Define the corner curvature κ C : C(G) R by κ C (v, f) = 1 v f. One immediately infers κ(v) = f F,v f κ C (v, f). This notion of curvature was first introduced in [BP01] and further studied in [BP06, Kel11] The Laplacians. Next, we recall the definition of the Laplacian in the special case of standard weights. Note that planar tessellations are special cases of these graphs. The general quadratic form for graphs with standard weights is given by Q : C(X) [0, ] Q(f) = 1 f(v) f(w) 2, 2 v w and as above we denote the space of functions f in C(X) such that Q(f) < by D. As discussed in Section , there are two canonical measures for graphs with standard weights. There is the counting measure which measures the volume of a set by counting the number the vertices. On the other hand, there is the degree measure deg which counts edges in a set W X. This can be seen by the identity deg(w ) = 2#E W + # W, where E W are the edges with both vertices in W and W are the edges having one vertex in W and one in X \ W. The identity above tells us that deg(w ) counts the edges with both end vertices in W twice and the edges leading out once. The counting measure gives rise to the Hilbert space l 2 (X) of complex valued functions whose square is summable. The scalar product on l 2 (X) is given by f, g = v X f(v)g(v), f, g l 2 (X), and the norm by f = f, f 1 2. By the discussion in Section the restriction Q to the subspace D l 2 (X) = {f l 2 (X) Q(f) < }. yields a closed positive quadratic form. By Theorem 1.14 we see that this form, denoted by Q (N) in Section , coincides with the form Q = Q (D) whose domain is the closure of the finitely supported functions C c (X) with respect to Q. Hence, the finitely supported functions are dense in the form domain.

59 3.2. CURVATURE AND THE BOTTOM OF THE SPECTRUM 57 Let be the positive selfadjoint operator associated to Q. Then, acts as f(v) = w v(f(v) f(w)) and by Theorem 1.14 it has the domain D( ) = {f l 2 (X) f l 2 (X)}. By Theorem 1.5 the operator is bounded if and only if sup v <. v X If one equips X with the degree measure deg, then the quadratic form Q restricted to the Hilbert space l 2 (X, deg) with scalar product f, g deg = v X f(v)g(v) deg(v), f, g l 2 (X, deg), is bounded by Theorem 1.5. The associated operator n, the normalized Laplacian, is then a bounded operator l 2 (X, deg) and acts as n f(v) = 1 (f(v) f(w)). deg(v) Recall that the subscript n stems from normalizing measure n which equals deg in the case of standard weights. w v 3.2. Curvature and the bottom of the spectrum In this section, we apply the general theory of the previous chapters to get explicit estimates for the bottom of the spectrum. First, we consider a lower bound that follows from an isoperimetric inequality and then an upper bound that follows from an estimate of the volume growth. The results of this section are proven in [KP11] Lower bounds. Recall the isoperimetric constant α introduced in Section and used in Section 1.5.5: α = inf W X finite # W deg(w ). In the case where the face degree is bounded by some q and the vertex degree is bounded by some p the following constant C p,q 1 will enter the estimate of the isoperimetric constant below 1 : if q =, C p,q := : if q < and p =, q 2 (1 + 2 )(1 + 2 ) : if p, q <. q 2 (p 2)(q 2) 2

60 58 3. CURVATURE ON PLANAR TESSELLATIONS Theorem 3.1 (Theorem 1 in [KP11]). Let G be a tessellation such that v p for all v X and f q for all f F with p, q [3, ]. Assume κ < 0 and let K := inf v X 1 κ(v). Then v α 2C p,q K. A key idea for the proof is a formula which is attributed in [BP01] to Harm Derksen which is state in the lemma below. It is an immediate consequence of the Gauß-Bonnet theorem and direct calculation. We call a set W V simply connected if W and V \ W are connected. Lemma 3.2. [BP01, Proposition 2.1.] Let W V be a finite simply connected subset of a planar tessellation. Then, v W κ(v) = 1 # W 2 + f F,f W,f (V \W ) #(f W ). f To get spectral estimates we need the following notation and observations. Let The inequality d = inf v X v and D = sup v. v X dλ 0 ( n ) λ 0 ( ) follows directly from the Rayleigh-Ritz characterization of the bottom of the spectrum, confer [Kel11]. Using this inequality together with Theorem 1.23 we obtain the following corollary from Theorem 3.1. Corollary 3.3. Let G be a tessellation such that v p for all v X and f q for all f F with p, q [3, ]. Assume κ < 0 and let K := inf v X 1 κ(v). Then v λ 0 ( n ) (1 1 4Cp,qK 2 2 ) 2K 2, and λ 0 ( ) d(1 1 4C 2 p,qk 2 ) 2dK 2, Remark. (a) The two inequalities on the right hand side in the corollary follow by the Taylor expansion of the square root and C p,q 1. (b) The theorem above can be considered as a discrete analogue to a theorem of McKean [McK70] who proves for an n-dimensional complete Riemannian manifold M with upper sectional curvature bound k that the bottom of the spectrum of the Laplace-Beltrami M 0 satisfies λ 0 ( M ) (n 1) 2 k/4.

61 3.2. CURVATURE AND THE BOTTOM OF THE SPECTRUM 59 (c) A fact noted by Higuchi [Hig01], see also [Żuk97], is that if κ < 0, then already κ 1/1806. This extremal case is assumed for a triangle, a heptagon and a 43-gon meeting in a vertex. This implies that if κ < 0, then K > 0 and, therefore, λ 0 ( n ) > 0 and λ 0 ( ) > 0. This recovers results of [Dod84, Hig01, Woe98] Upper bounds. In this section we discuss volume growth bounds for tessellations whose face degree is constantly q. We call such tessellations q-face regular. As a consequence, this yields upper bounds for the bottom of the essential spectrum. Denote by S r the vertices with combinatorial graph distance r 0 to a center vertex o. We will suppress the dependence on o in notation since it is not important for connected graphs. Furthermore, let B r = r k=0 S k. We use the upper exponential volume growth µ = µ n defined in Section : µ = lim sup n 1 r log #B r. The result will be stated in terms of normalized average curvatures over spheres ( ) 2q 1 κ r := κ(s r ) := κ(s r ). q 2 #S r Note that the constant 2π(q 2)/2q is the internal angle of a regular q-gon. First, we present a volume growth comparison theorem which is an analogue to the Bishop-Guenther-Gromov comparison theorem from the Riemannian setting. Theorem 3.4 (Theorem 3 in [KP11]). Let G = (X, E, F ) and G = ( X, Ẽ, F ) be two q-face regular tessellations with non-positive vertex curvature, S r X and S r X be spheres with respect to the centers o X and õ X, respectively. Assume that the normalized average spherical curvatures satisfy κ( S r ) κ(s r ) 0, r 0. Then the difference sequence (# S r #S r ) satisfies # S r #S r 0, is monotone non-decreasing and, in particular, we have µ( G) µ(g). We furthermore get an explicit recursion formula for the growth in terms of the normalized average spherical curvatures. This result can be proven for tessellations without cut locus. That is for every v the distance function d(, v) has no local maxima. For example, this is implied by non-positive corner curvature [BP06, Theorem 1]. In our case of face regular graphs, non-positive corner curvature is equivalent

62 60 3. CURVATURE ON PLANAR TESSELLATIONS to non-positive curvature. However, the theorem below is not restricted to the non-positive curvature case. For 3 q < let N = q 2 if q is even and N = q 2 if q is odd, 2 and b l = for 0 l N 1. { 4 q q 2 N 1 : if q is odd and l = : else, Theorem 3.5 (Theorem 2 in [KP11]). Let G = (X, E, F ) be a q- face regular tessellation without cut locus. Then we have the following (N + 1)-step recursion formulas for r 1 r 1 l=0 (b l κ(s r l ))#S r l + #S 1 : if r < N, #S r+1 = N 1 l=0 (b l κ(s N l ))#S N l : if r = N, N 1 l=0 (b l κ(s r l ))#S r l #S r N : if r > N. Idea of the proof. The proof given in [KP11] uses strongly the assumption of constant face degree. We count the number of faces c r j that intersect a ball B r in j vertices, where 1 j q and q is the constant face degree. If j q 2, then c r j is equal to the number of faces c r+1 j+2 that intersect the ball B r+1 in j + 2 vertices. Finally, one has to relate the numbers c r j to the number of vertices in a sphere S r. The (N + 1)-step recursion formula in the theorem above gives rise to a recursion matrix M r, r 0, mapping R N to R N such that M r (#S r N,..., #S r ) = (#S r N+1,..., #S r+1 ). In the special case when also the vertex degree is constant, say p, we have a (p, q)-regular tessellation. Then, the constant b l κ k is equal to p 2, except for l = (N 1)/2 and q odd. In particular, there is a matrix M such that M = M r, r 0. The characteristic polynomial of M is then given by the complex polynomial if q is even, and 2, g p,q (z) = 1 (p 2)z (p 2)z N + z N+1, g p,q (z) = 1 (p 2)z (p 4)z N+1 2 (p 2)z N + z N+1, if q is odd. By [CW92] and [BCS02], g p,q is a reciprocal Salem polynomial, i.e., its roots lie on the complex unit circle except for two positive reciprocal real zeros This yields 1 x p,q < 1 < x p,q < p 1. µ = log x p,q

63 3.3. DECREASING CURVATURE AND DISCRETE SPECTRUM 61 in the special case of (p, q)-regular tessellation. In particular, the considerations above recover the results of Cannon and Wagreich [CW92] and Floyd and Plotnick [FP87, Section 3] that the growth function is a rational function. Now, we combine these insights with the discrete version of Brook s theorem by Fujiwara [Fuj96a] and the observation λ ess 0 ( n ) 1 2eµn/2 e µn + 1 λ ess 0 ( ) D λ ess 0 ( n ) with D = sup K X finite inf v X\K v, to get the following estimate on the bottom of the essential spectrum of n and. Theorem 3.6. Let G be a q-face regular tessellation such that ( 1 κ(v) p p ) 0, v X, q for some integer p 3. Then and 0 ( n ) 1 2x1/2 p,q x p,q + 1 λ ess ( λ ess 0 ( ) D 1 2x1/2 p,q x p,q + 1 where x p,q is the largest real zero of g p,q above Decreasing curvature and discrete spectrum In this section, we study the case of uniformly decreasing curvature. More precisely, we look at tessellations where κ = inf sup K Xfinite v X\K κ(v) equals. For this case, we discuss that the spectrum of n is discrete except for the point 1 and the spectrum of consists only of discrete eigenvalues which accumulate at. In this case, we denote the eigenvalues of in increasing order counted with multiplicity by λ j ( ), j Discrete spectrum. First, we address the spectrum of n. As a bounded operator, n has non empty essential spectrum. In [Kel10, Theorem 5] it was discussed that if the essential spectrum of n consists of one point then this point must be 1. Theorem 3.7 (Theorem 3 (a) in [Kel10]). Let G be a tessellation. The essential spectrum of n consists only of the point 1 if κ =. ),

64 62 3. CURVATURE ON PLANAR TESSELLATIONS It can be seen by examples that the converse implication does not hold in general. As the operator is unbounded, it may have empty essential spectrum. The next theorem characterize this case. Theorem 3.8 (Theorem 3 (b) in [Kel10]). Let G be a tessellation. The spectrum of is purely discrete if and only if κ =. Idea of the proof. If the spectrum of is purely discrete, then by the Perrson type theorem ϕ n, ϕ n for every normalized sequence (ϕ n ) in l 2 (V ) converges weakly to zero. For a sequence (v n ) of vertices and the delta functions δ vn, one finds δ vn, δ vn = v n 2κ(v n ). This implies κ =. Now, assume κ =. By Theorem 3.1 we infer that outside of large enough finite sets the isoperimetric constant is uniformly positive. Moreover, by results as Theorem 1.23 the bottom of the spectrum of the operator restricted to functions supported outside of larger and larger finite sets converges to. This, however, is equivalent to pure discrete spectrum, as the restricted operators are finite rank perturbations of the original operator. Remark. (a) The theorems above can be considered as discrete analogues of a theorem of Donnelly/Li [DL79]. This theorem states that, for a negatively curved, complete Riemannian manifold M with sectional curvature bound decaying uniformly to, the Laplace- Beltrami operator M has purely discrete spectrum. (b) In [Fuj96b] Fujiwara proved the statement of Theorem 3.7 for the normalized Laplacian n on trees. (c) Wojciechowski [Woj08] showed discreteness of the spectrum of on general graphs with standard weights in terms of a different quantity which is sometimes referred to as a mean curvature (see also the discussion in Section 1.4) Eigenvalue asymptotics. An important observation in the proof of the theorem above is the following estimate v v κ(v) 1 2 6, v X. That implies that and κ go simultaneously to. In particular, if κ =, then there is a bijective map N 0 X, j v j, such that v j v j+1, j 0. In [BGK13] it was observed that planar graphs are sparse. Hence, the results of Section can be used to obtain the following eigenvalue asymptotics.

65 3.5. CURVATURE ON PLANAR GRAPHS 63 Theorem 3.9. Let G be a tessellation. If κ =, then λ j ( ) lim j v j = The l p spectrum Now, we turn to the spectrum of the Laplacians as operators on l p (X, deg) and l p (X), p [1, ]. For the normalized Laplacian n, consider the generalized Laplacian L n on C(X) with the same mapping rule. By Theorem 1.5, we find that the restriction (p) n of L n to l p (X, deg), p [1, ] is a bounded operator. It can easily be seen that (p) n coincides with the generator of the extension of the semigroups e t n to l p (X, deg), p [1, ) and ( ) n being the adjoint of (n) n. Simultaneously, let L be the generalized Laplacian as is an extension of on C(X). Then, it can be seen by Theorem 1.13 that the restriction (p) of L to D( (p) ) = {f l p (X) f l p (X)} is the generator of the extension of the semigroup e t to l p (X), p [1, ), and ( ) is the adjoint of (1). A famous question brought up by Simon [Sim80] and answered by Hempel/Voigt [HV86] for Schrödinger operators is whether the spectrum depends on the underlying Banach space. Sturm, [Stu93], addressed this question for uniformly elliptic operators on manifolds in terms of uniform subexponential volume growth. As a special case, he considers curvature bounds. We already discussed the analogue of the general result of Sturm obtained in [BHK13] in Section 2.6. As a consequence of this theorem and some geometric and functional analytic ingredients, one can derive the following theorem which is taken from in [BHK13]. Theorem (a) If κ 0, then σ( (2) ) = σ( (p) ) for p [1, ]. (b) If K κ < 0, then λ 0 ( (2) ) λ 0 ( (1) ). (c) If κ =, then σ( (2) ) = σ( (p) ) for all p (1, ) Curvature on planar graphs We close this thesis by some considerations on curvature for general planar graphs. This was investigated in [Kel11]. For a general planar graph, we have to extend the definitions of degrees of faces and vertices. For a corner (v, f) C(G), we denote by (v, f) the minimal number of times the vertex v is met by a boundary walk of f. Then, we define, for v X and f F, v = (v, g) and f = (w, f). (v,g) C(G) (w,f) C(G)

66 64 3. CURVATURE ON PLANAR TESSELLATIONS As the degree of corners in a tessellation is always one, these definitions coincide with the ones above in the case of tessellations. We say a face f is unbounded if f =. A graph is called simple if f 3 for all f F. (By the way we defined graphs in this thesis, this is always satisfied since b(x, x) = 0 and b(x, y) 1.) With this notation, we define the corner curvature κ C : C(G) R by κ C (v, f) = 1 v f and the vertex curvature by κ : X R by κ(v) = (v, f) κ C (v, f). (v,f) C(G) These definitions are consistent with the definition of κ C and κ on tessellations and they also satisfy a Gauß-Bonnet formula [Kel11, Proposition 1]. Next, we look at a generalization of tessellations. We call a face a polygon if it is homeomorphic to the open unit disc and we call it an infinigon if it is homeomorphic to the upper half space in R 2. A planar graph with a locally compact embedding is called locally tessellating if it satisfies the following conditions: (T1) Every edge is contained in two faces. (T2*) Two faces are either disjoint or intersect in a vertex or in a path of edges. If this path consists of more than one edge, then both faces are unbounded. (T3*) Every face is a polygon or an infinigon. Here, (T1) is the same as in the tessellation case. This class of graphs includes tessellations and trees as well as hybrids. In [Kel11] we find the following theorem which shows that non-positive curvature on planar graphs implies that the graph is almost a tessellation, i.e., it is locally tessellating. Theorem 3.11 (Theorem 1 in [Kel11]). Let G be a connected, locally finite, planar graph. If κ C 0 or if G is simple with κ 0 then G is locally tessellating and infinite. For the proof we assume the contrary. One isolates finite areas of the graphs on which the assumptions (T1), (T2*), (T3*) fail. Such an area is then copied finitely many times and pasted along its boundary to be finally embedded into the 2-dimensional unit sphere. Here, the Gauß-Bonnet theorem is used to show that there must be some positive curvature. By taking enough copies this positive curvature can not come from the vertices where we pasted the graph but from the inside. Furthermore, in [Kel11, Theorem 2] it is shown that locally tessellating graphs can be embedded into tessellations in a suitable way.

67 3.5. CURVATURE ON PLANAR GRAPHS 65 This way one can carry over results from tessellations to locally tessellating graphs and by the theorem above to planar graphs in the case of non-positive curvature. Among the geometric applications in the paper are the following some of which are extensions of [BP01, BP06]: Absence of cut locus, i.e., every distance minimizing path can be continued to infinity [Kel11, Theorem 3]. A description of the boundary of distance balls [Kel11, Theorem 4]. Bounds for the growth of distance balls [Kel11, Theorem 5]. Positivity and bounds for an isoperimetric constant constant [Kel11, Theorem 6]. Empty interior for minimal bigons and Gromov hyperbolicity [Kel11, Theorem 7]. The first two results are obtained for non-positive curvature and the other three for negative curvature. Furthermore, there are applications to the spectral theory. Let us mention that the isoperimetric estimates mentioned above yield analogues to the results in Section Simultaneously, the results of Section 3.3 carry over by the virtue of [Kel11, Theorem 2]. Let us close this section by a result on absence of compactly supported eigenfunctions. For tessellations such a result was proven in [KLPS06]. In [Kel11] a simplified proof is given in the more general setting of planar graphs (which are locally tessellating in the case of non-positive curvature by what we discussed above). Theorem 3.12 (Theorem 9 in [Kel11]). Let G be a connected, locally finite, planar graph such that κ C 0. Then, neither nor n admit finitely supported eigenfunctions. While such a result is true in great generality in continuous settings, it can easily be seen that it may even fail when only κ 0 (or even κ < 0) is assumed, [BP06, KPP].

68

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75 Part 2 Original Manuscripts

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77 CHAPTER 4 M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, Journal für die reine und angewandte Mathematik 2012 (2012),

78 J. reine angew. Math., Ahead of Print DOI /CRELLE Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin New York 2011 Dirichlet forms and stochastic completeness of graphs and subgraphs By Matthias Keller and Daniel Lenz at Jena Abstract. We study Laplacians on graphs and networks via regular Dirichlet forms. We give a su cient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all l p,1ep < y, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. Introduction There is a long history to the study of the heat equation and spectral theory on graphs and networks (see e.g. the monographs [4], [5] and references therein). The corresponding operators are known as Laplacians, acoustic operators or generators of symmetric Markov processes on the graph or network. A substantial part of this literature is devoted to graphs giving Laplacians, which are bounded on l 2. Recently, certain basic questions concerning unbounded Laplacians have received attention. This is the starting point for our paper. More precisely, we use the framework of regular Dirichlet forms in order to define the Laplacians on networks via forms (Section 1), study essential selfadjointness (Theorem 6), determine the generators of the associated semigroups on l p,1e p < y, under suitable conditions (Theorem 5), characterize stochastic completeness (Theorem 1), investigate the relationship between stochastic completeness of graphs and subgraphs (Theorem 2, Theorem 3, Theorem 4). The use of Dirichlet forms allows us to deal with these questions in a rather general setting. In particular, our results seem to extend all earlier corresponding results. Furthermore, we

79 2 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs hope that our results and the thorough discussion of background and context may be useful in the study of further questions as well. Let us discuss these topics in more detail: There are recent investigations of essential selfadjointness of corresponding Laplacians by Jorgensen [17], of stochastic completeness by Dodziuk and Matthai [11], and of both essential selfadjointness and stochastic completeness by Dodziuk [9], Wojciechowski [28] (see [29] as well) and Weber [26]. These investigations deal with locally finite graphs and the associated operators. While [11], [9] treat bounded Laplacians, [17], [28], [26] do neither assume a uniform bound on the vertex degree nor a modification of the measure and, accordingly, the resulting Laplacians are not necessarily bounded. It turns out that all the Laplacians in question are special instances of generators of regular Dirichlet forms on discrete sets. In fact, there is a one-to-one correspondence between the regular Dirichlet forms on a discrete set and graphs over this set with weights satisfying a certain summability condition. This naturally raises the question to which extent similar results to the ones in [9], [11], [17], [26], [28] also hold for arbitrary regular Dirichlet forms on discrete sets. Our first result, Theorem 1, characterizes stochastic completeness for all regular Dirichlet forms on discrete sets. This generalizes a main result of [28] (see [9], [11], [26] as well for related results and a su cient condition for stochastic completeness), which in turn is inspired by Grigor yan s corresponding result for manifolds [15]. Of course, in terms of methods our considerations concerning stochastic completeness heavily draw on existing literature as e.g. Sturm s [23] for strongly local Dirichlet forms and Grigor yans results [15] on Riemannian manifolds. A crucial di erence, however, is that our Dirichlet forms are not local. In this sense our results can be understood as providing some non-local counterpart to [23], [15]. It should be emphasized that unlike the cited literature we do allow for non vanishing killing terms. In order to make sense out of a notion of stochastic completeness in the presence of a killing term we actually have to extend the usual definition. This is done by our concept of stochastic completeness at infinity ðsc y Þ and stochastic incompleteness at infinity ðsi y Þ. Let us be a bit more precise: Stochastic completeness concerns loss or conservation of heat. Now, loss of heat may occur for two reasons. One reason is killing within the graph by non-vanishing killing term. The other reason is heat transport to infinity or the boundary in finite time. This transport to infinity may happen irrespective of presence of a killing term. It is this transport to infinity which is captured by our notion of stochastic completeness at infinity. Of course, in the case of vanishing killing term stochastic completeness and stochastic completeness at infinity agree. Our Theorem 1 gives a unified treatment of the situation. Note that strengthening of the killing may make the graph actually more complete at infinity as discussed in Theorem 2. Let us also mention strongly related work of Feller [12], [13] and of Reuter [22] dealing with uniqueness of Markov process on discrete sets with given weights. While these works use di erent methods and seem to have been somewhat neglected in the above mentioned literature, they in fact cover parts of the abstract results on stochastic completeness discussed in [28], [26]. They are in some sense even more general in that they do not assume symmetry of the Markov process. We will discuss this more specifically after the statement of our corresponding result. However, we stress already here that a crucial part of our result is not covered by [12], [22] as we allow for both a killing term and for arbitrary measures on our underlying set.

80 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 3 Let us emphasize that our treatment requires intrinsically more e ort than the considerations of [9], [11], [26], [28] as in our setting the Laplacians (i.e., generators of the Dirichlet forms) are known much less explicitly. In fact, in general not even the functions with compact support will be in the domain of definition of our Laplacians. As the functions with compact support need not belong to the domain of definition of our Laplacians, the question of essential selfadjointness does in general not make sense in our context. On the other hand, if the functions with compact support belong to the domain of definition and a certain geometric condition called ðaþ below is satisfied, we can prove essential selfadjointness of the Laplacians in question on the set of functions with compact support (Theorem 6). This result extends the corresponding result of [17], [9], [26], [28] to all regular Dirichlet forms on discrete sets. Note that this (again) includes the presence of an arbitrary killing term and an arbitrary measure on our discrete set. We also give examples in which essential selfadjointness fails (as does condition ðaþ). Along our way, we can also determine the generators for the corresponding semigroups on all l p, p A ½1; yþ, for all regular Dirichlet forms on graphs satisfying ðaþ. These generators turn out to be the maximal ones (Theorem 5). These results seem to be new even in the situations considered in [12], [9], [11], [17], [22], [26], [28]. After these considerations, our final aim is to study how ðsc y Þ of a subgraph is related to ðsc y Þ of the whole graph. There, we obtain two results: We show that any graph is a subgraph of a graph satisfying ðsc y Þ. This completion can be achieved both by adding killing terms (Theorem 2) and by adding edges (Theorem 3). We also show that stochastic incompleteness of a suitably modified subgraph implies stochastic incompleteness of the whole graph (Theorem 4). These results seem to be new even in the contexts discussed earlier. We have tried to make this paper as accessible and self-contained as possible for both people with a background in Dirichlet forms and people with a background in geometry. For this reason some arguments are given, which are certainly well known. For further studies of certain spectral features of Laplacians in the framework developed below we refer the reader to [16], [19], both of which were written after the present paper. The paper is organized as follows. In Section 1 we present the notation and our main results. A study of basic properties of Dirichlet forms on graphs can be found in Section 2. In Section 3 we consider Dirichlet forms on graphs satisfying the condition ðaþ mentioned above. For these forms we calculate the generators of the l p semigroups for p A ½1; yþ and we show essential selfadjointness of the generators on l 2 (whenever the functions with compact support are in the domain of definition). In Section 4 we give examples where essential selfadjointness fails as well as examples of non-regular Dirichlet forms on graphs. A short discussion of the heat equation in our framework is given in Section 5. Section 6 deals with extending the semigroup and resolvent in question to a somewhat larger space of functions. In Section 7 we can then prove our result characterizing stochastic completeness for arbitrary Dirichlet forms on graphs. Section 8 contains a proof that any graph is a subgraph of a stochastically complete graph and that any graph can be made stochastically complete by adding a killing term. Section 9 contains an incompleteness criterion.

81 4 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 1. Framework and results Throughout V will be a countable set. Let m be a measure on V with full support (i.e. m is a map m : V!ð0; yþ). Then, ðv; mþ is a measure space. We will deal exclusively with real valued functions. Thus, l p ðv; mþ, 1e p < y, is defined by nu : V! R : P x A V o mðxþjuðxþj p < y : Obviously, l 2 ðv; mþ is a Hilbert space with inner product given by hu; vi :¼ P mðxþuðxþvðxþ and norm kuk :¼ hu; ui 1 2 : x A V Moreover we denote by l y ðvþ the space of bounded functions on V. Note that this space does not depend on the choice of m. It is equipped with the supremum norm kk y. A symmetric non-negative form on ðv; mþ is given by a dense subspace D of l 2 ðv; mþ called the domain of the form and a map Q : D D! R with Qðu; vþ ¼Qðv; uþ and Qðu; uþ f 0 for all u; v A D. Such a map is already determined by its values on the diagonal. For u A l 2 ðv; mþ we then define QðuÞ by QðuÞ :¼ Qðu; uþ if u A D and QðuÞ :¼ y otherwise. If l 2 ðv; mþ!½0; yš, u 7! QðuÞ, is lower semicontinuous, Q is called closed. If Q has a closed extension, it is called closable and the smallest closed extension is called the closure of Q. A map C : R! R with Cð0Þ ¼0 and jcðxþ CðyÞj e jx yj is called a normal contraction. If Q is both closed and satisfies QðCuÞ e QðuÞ for all u A l 2 ðv; mþ and all normal contractions C, it is called a Dirichlet form on ðv; mþ (see [3], [6], [14], [20] for background on Dirichlet forms). Let C c ðvþ be the space of finitely supported functions on V. A Dirichlet Q form on ðv; mþ is called regular if DðQÞ X C c ðvþ is both dense in C c ðvþ with respect to the supremum qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi norm and dense in DðQÞ with respect to the form norm given by kk Q :¼ kk 2 þ QðÞ. As discussed below, for such a regular form the set C c ðvþ is necessarily contained in the form domain. Thus, a Dirichlet form Q on ðv; mþ is regular if and only if it is the closure of its restriction to the subspace C c ðvþ. Regular Dirichlet forms on ðv; mþ are given by graphs on V, as we discuss next (see Section 2 for details). A symmetric weighted graph over V or a symmetric Markov chain on V is a pair ðb; cþ consisting of a map b : V V!½0; yþ with bðx; xþ ¼0 for all x A V and a map c : V!½0; yþ satisfying the following two properties: (b1) bðx; yþ ¼bðy; xþ for all x; y A V. (b2) P y A V bðx; yþ < y for all x A V.

82 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 5 We can then think of ðb; cþ or rather the triple ðv; b; cþ as a weighted graph with vertex set V in the following way: An x A V with cðxþ 3 0 is then thought to be connected to the point y by an edge with weight cðxþ. Moreover, x; y A V with bðx; yþ > 0 are thought to be connected by an edge with weight bðx; yþ. The map b is called the edge weight. The map c is called killing term. Vertices x; y A V with bðx; yþ > 0 are called neighbors. More generally, x; y A V are called connected if there exist x 0 ; x 1 ;...; x n ; x nþ1 A V with bðx i ; x iþ1 Þ > 0, i ¼ 0;...; n, and x 0 ¼ x, x nþ1 ¼ y. This allows us to define connected components of V in the obvious way. To ðv; b; cþ we associate the Q comp : C c ðvþ!½0; yþ given by form Q comp ¼ Q comp b; c on C c ðvþ with diagonal Q comp ðuþ ¼ 1 2 P x; y A V bðx; yþ uðxþ uðyþ 2 þ P x A V cðxþuðxþ 2 : Obviously, Q comp diagonal given by is a restriction of the form Q max ¼ Q max b; c; m defined on l2 ðv; mþ with Q max ðuþ ¼ 1 2 P x; y A V bðx; yþ uðxþ uðyþ 2 þ P x A V cðxþuðxþ 2 : Here, the value y is allowed. It is not hard to see that Q max is closed and hence Q comp is closable on l 2 ðv; mþ (see Section 2) and the closure will be denoted by Q ¼ Q b; c; m and its domain by DðQÞ which is the closure of C c ðvþ with respect to kk Q. Then, there exists a unique selfadjoint operator L ¼ L b; c; m on l 2 ðv; mþ such that and DðQÞ ¼Domain of definition of L 1=2 QðuÞ ¼hL 1=2 u; L 1=2 ui for u A DðQÞ (see e.g. [6], Theorem 1.2.1). As Q is non-negative so is L. Moreover, it is not hard to see that Q max ðcuþ e Q max ðuþ for all u A l 2 ðv; mþ (and in fact any function u) and every normal contraction C. Theorem 3:1:1 of [14] then implies that Q also satisfies QðCuÞ e QðuÞ for all u A l 2 ðv; mþ and hence is a Dirichlet form. By construction it is regular. In fact, every regular Dirichlet form on ðv; mþ is of the form Q ¼ Q b; c; m (see Theorem 7 in Section 2). Remark. Our setting generalizes the setting of [9], [11], [17], [26], [28] to Dirichlet forms on countable sets. In our notation, the situation of [11], [26], [28] can be described by the assumptions m 1 1, c 1 0, and bðx; yþ A f0; 1g for all x; y A V with x 3 y and the setting of [9], [17] can be described by m 1 1, c 1 0 and bðx; yþ ¼0 for all but finitely many y for each x A V. In particular, unlike [9], [11], [17], [26], [28] we do not assume finiteness of the sets fy A V : bðx; yþ > 0g for all x A V. Let now a measure m on V with full support and a weighted graph ðb; cþ over V be given. Let Q be the associated form and L its generator. Then, by standard theory [7], [14], [20], the operators of the associated semigroup e tl, t f 0, and the associated resolvent aðl þ aþ 1, a > 0, are positivity preserving and even markovian.

83 6 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs Positivity preserving means that they map non-negative functions to non-negative functions. In fact, if ðv; b; cþ is connected they are even positivity improving, i.e., map non-negative nontrivial functions to positive functions (see Section 2). Markovian means that they map non-negative functions bounded by one to non-negative functions bounded by one. This can be used to show that semigroup and resolvent extend to all l p ðv; mþ, 1 e p e y. These extensions are consistent, i.e., two of them agree on their common domain and they are selfadjoint, i.e., the adjoint to the extension to l p ðv; mþ with 1 e p < y is given by the extension to l q ðv; mþ for 1=p þ 1=q ¼ 1, see [6]. The corresponding generators are denoted by L p. Thus, the extension of ðl þ aþ 1 to l p ðv; mþ is given by ðl p þ aþ 1. In particular we have L ¼ L 2. We can describe the action of the operator L p explicitly (in Section 2) as follows (see Theorem 9): Define the formal Laplacian ~L ¼ ~L b; c; m on the vector space ð1þ ~F :¼ n u : V! R : P y o jbðx; yþuðyþj < y for all x A V by ~LuðxÞ :¼ 1 mðxþ P y bðx; yþ uðxþ uðyþ þ cðxþ mðxþ uðxþ; where, for each x A V, the sum exists by assumption on u. Then, L p is a restriction of ~L for any p A ½1; yš. After having discussed the fact that these are di erent semigroups on di erent l p spaces, we will now follow the custom and write e tl for all of them. The preceding considerations show that 0 e e tl 1ðxÞ e 1 for all t f 0 and x A V. The question, whether the second inequality is actually an equality has received quite some attention. In the case of vanishing killing term, this is discussed under the name of stochastic completeness or conservativeness. In fact, for c 1 0 and bðx; yþ A f0; 1g for all x; y A V, there is a characterization of stochastic completeness of Wojciechowski [28] (see [9], [11], [26] for related results as well). This characterization is an analogue to corresponding results for Markov processes [12], [22], results on manifolds of Grigor yan [15] and results of Sturm for general strongly local Dirichlet forms [23]. Our first main result concerns a version of this result for arbitrary regular Dirichlet forms on weighted graphs (see Section 7 for details and proofs concerning the subsequent discussion): In this case, we have to replace e tl 1 by the function M t ðxþ :¼ e tl 1ðxÞþ Ðt e sl c ðxþ ds; x A V: m 0

84 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 7 This is well defined, satisfies 0 e M e 1 and for each x A V, the function t 7! M t ðxþ is continuous and even di erentiable. Note that for c 1 0, we obtain M ¼ e tl 1 whereas for c 3 0 we obtain M t > e tl 1 on any connected component of V on which c does not vanish identically (as the semigroup is positivity improving). The term e tl 1 can be interpreted as the amount of heat contained in the graph at time t and the integral can be interpreted as the amount of heat killed within the graph up to the time t. Thus, 1 M t is the amount of heat transported to the boundary of the graph by the time t and M t can be interpreted as the amount of heat, which has not been transported to the boundary of the graph at time t. Our question then becomes whether the quantity 1 M t vanishes identically or not. Our result then reads as follows. Theorem 1 (Characterization of heat transfer to the boundary). Let ðv; b; cþ be a weighted graph and m a measure on V of full support. Then, for any a > 0, the function w :¼ Ðy 0 ae ta ð1 M t Þ dt satisfies 0 e w e 1, solves ð ~L þ aþw ¼ 0, and is the largest non-negative l e 1 with ð~l þ aþl e 0. In particular, the following assertions are equivalent: (i) For any a > 0 there exists a nontrivial, non-negative bounded l with ð~l þ aþl e 0. (ii) For any a > 0 there exists a nontrivial bounded l with ð~l þ aþl ¼ 0. (iii) For any a > 0 there exists a nontrivial, non-negative bounded l with ð~l þ aþl ¼ 0. (iv) The function w is nontrivial. (v) M t ðxþ < 1 for some x A V and some t > 0. (vi) There exists a nontrivial bounded non-negative N : V ½0; yþ!½0; yþ satisfying ~LN þ d dt N ¼ 0 and N Remark. (a) Conditions (ii) and (iii) deal with eigenvalues of ~L considered as an operator on l y ðvþ. In particular, (ii) must fail (for su ciently large a) whenever ~L gives rise to a bounded operator on l y ðvþ. Thus, any bounded operator ~L yields a stochastically complete graph. In this way we recover the corresponding results of [9], [11]. (b) The case c 1 0, m 1 1, bðx; yþ A f0; 1g recovers the corresponding result of [28]. In fact, in the case c 1 0, m 1 1 and general (not even symmetric) b the equivalence of (i) (or (ii)) and (v) is already discussed in [12], [22]. These works mainly aim at studying uniqueness of the Markov process, i.e., a (somewhat weaker) version of (vi). They characterize this uniqueness by validity of (i) for m 1 1 and arbitrary c. In this sense it seems fair to say that for c 1 0 the equivalence of (i), (v) and (vi) is well known and for general c the

85 8 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs equivalence of (i) and (vi) is well known. Besides our new proof (inspired by [28], [15]), our main contribution here is the definition of M allowing for an extension of (v) to situations with killing terms. (c) The minimum principle discussed below, Theorem 8, will show that for a > 0 any nontrivial, non-negative solution u of ð~l þ aþu ¼ 0 satisfies u > 0 if the underlying graph is connected. (d) Let L be the operator associated to a weighted graph ðv; b; cþ and L 0 the operator associated to ðv; b; 0Þ, both with respect to the same measure m : V!ð0; yþ. The equivalence of (i) and (v) in the theorem above obviously implies M t ¼ 1 whenever e tl 0 1 ¼ 1, since ~Ll f ~L 0 l for every non-negative l A ~F. The previous theorem suggests the following definition for stochastic completeness at infinity and stochastic incompleteness at infinity for general Dirichlet forms on weighted graphs. Definition 1.1. The weighted graph ðv; b; cþ with the measure m of full support is said to satisfy ðsi y Þ if it satisfies one (and thus all) of the equivalent assertions of Theorem 1. Otherwise ðv; b; cþ is said to satisfy ðsc y Þ. Remark. Note that validity of ðsi y Þ depends on both ðv; b; cþ and m. In fact, for given ðv; b; cþ it is always possible to choose m in such a way that ~L becomes a bounded operator on l y ðvþ. Then, ðsc y Þ holds (by (a) of the previous remark). The following two results show how graphs can be made to satisfy ðsc y Þ by addition of killing terms or edges. They seem to be new even in the setting considered in [9], [11], [26], [28]. Theorem 2. Let m be a measure on V with full support. For any weighted graph ðv; b; cþ there is c 0 : V!½0; yþ such that ðv; b; c þ c 0 Þ satisfies ðsc y Þ. Remark. Of course, addition of killing terms yields to loss of mass from the graph reflected in the inequality e tl 1 < 1. As our concept of ðsc y Þ only considers mass transported to the geometric boundary of the graph, we can have and even enforce ðsc y Þ by adding killing terms. More precisely, the theorem can be understood in the following way: Adding a killing term kills heat within the graph on any vertex where the killing term does not vanish. If we eliminate enough heat by the killing terms, we can achieve that no more heat is transferred to the geometric boundary of the graph. A subgraph ðw; b W ; c W Þ of a weighted graph ðv; b; cþ is given by a subset W of V and the restriction b W of b to W W and the restriction c W of c to W. The weighted graph ðv; b; cþ is then called a supergraph to ðw; b W ; c W Þ. Given a measure m on V we denote its restriction to W by m W. The subgraph ðw; b W ; c W Þ then gives rise to a form on l 2 ðw; m W Þ with associated operator L bw ; c W ; m W. Theorem 3. Any weighted graph is the subgraph of a weighted graph satisfying ðsc y Þ. This supergraph can be chosen to have vanishing killing term if the original graph has vanishing killing term.

86 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 9 Remark. Note that (in the common definitions) the volume growth of balls in a graph dominates the volume growth of balls in any of its subgraph. Thus, the theorem has the consequence that failure of ðsc y Þ can not be inferred from lower bounds on the growth of volumes of balls. While subgraphs do not force incompleteness according to Theorem 3, suitably adjusted subgraphs do force incompleteness of the whole graph. In order to be more precise, we need some more notation. Let ðv; b; cþ be a weighted graph with measure m of full support and W a subset of V. Let m W be the restriction of m to W. Let i W : l 2 ðw; m W Þ!l 2 ðv; mþ be the canonical embedding, i.e., i W ðuþ is the extension of u to V by setting i W ðuþ identically zero outside of W. Let p W : l 2 ðv; mþ!l 2 ðw; m W Þ be the canonical projection, i.e., the adjoint comp; ðdþ of i W. Then, W gives rise to the form Q defined on C c ðwþ by Here, d W ðxþ :¼ Q comp; ðdþ W P y A VnW the same vertex in V. Thus, Q ðw; b ðdþ W ; cðdþ W Þ with W ðuþ :¼ Qði W uþ¼q comp b W ; c W ðuþþ P x A W d W ðxþuðxþ 2 : bðx; yþ describes the edge deficiency of vertices in W compared to comp; ðdþ W is in fact the form Q comp of the weighted graph b ðdþ W ¼ b W and c ðdþ W ¼ c W þ d W : In particular, by the theory developed above, its closure in l 2 ðw; m W Þ, denoted by Q ðdþ W, is a Dirichlet form. The associated selfadjoint operator will be denoted by L ðdþ W. This operator is sometimes thought of as a restriction of the original operator to W with Dirichlet boundary condition. For this reason we include the superscript D in the notation. Another interpretation (suggested by the above expression for the form) is to think about the graph which arises from the subgraph W by adding one way edges to a vertex at infinity according to the mentioned edge deficiency. Again, it is not hard to express the action of L ðdþ W explicitly. In fact, the above considerations applied to the graph ðw; b ðdþ W ; cðdþ W ; m W Þ show that L ðdþ W u ¼ ~L ðdþ W u for any u A DðL ðdþ W Þ. Here, the formal Dirichlet Laplacian ~L ðdþ W p W ~F ¼ iw 1ð ~FÞ and given by on W is defined on ~L ðdþ W uðxþ ¼ 1 P b ðdþ W mðxþ ðx; yþ uðxþ uðyþ þ c ðdþ W ðxþuðxþ ¼ ~Li W uðxþ y A W for x A W. These considerations give that for a function u on W (which is extended by 0 to V) the equality ð2þ ~L ðdþ W uðxþ ¼ ~LuðxÞ

87 10 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs holds for any x A W. This will be used repeatedly in the sequel. Note also that for W ¼ V we recover the operator on the whole graph, i.e., ~L ðdþ V ¼ ~L and L ðdþ V ¼ L. [28]. The following result seems to be new even in the setting considered in [9], [11], [26], Theorem 4. Let ðb; cþ be a weighted graph over V and m a measure on V of full support. Then ðsi y Þ holds, whenever there exists W L V such that the weighted graph ðb ðdþ W ; cðdþ W Þ over the measure space ðw; m W Þ satisfies ðsi y Þ. As an example of a situation in which the theorem may be applied we note the following consequence. Corollary 1.2. Let ðb; 0Þ be a weighted graph over V with vanishing killing term and m a measure on V of full support. Let W be a subset of V such P that ðb ðdþ W ; 0Þ over the measure space ðw; m W Þ satisfies ðsi y Þ and there exists C > 0 with bðx; yþ=mðxþ e C for any x A W. Then ðb; 0Þ over ðv; mþ satisfies ðsi y A VnW y Þ. So far, we have not discussed the precise domains of definition for our operators. In fact, the actual domains have been quite irrelevant for our considerations. To determine the domains we need a geometric condition saying that any infinite path has infinite measure. More precisely, we define condition ðaþ as follows: (A) The equality P mðx n Þ¼y holds for any sequence ðx n Þ of elements of V such n A N that bðx n ; x nþ1 Þ > 0 for all n A N. Of course, an equivalent requirement would be that the equality mðfx n : n A NgÞ ¼ y holds for any sequence ðx n Þ of pairwise di erent elements of V such that bðx n ; x nþ1 Þ > 0 for all n A N. Note that ðaþ is a condition on ðv; mþ and ðb; cþ together. If inf x A V mðxþ > 0 holds, then ðaþ is satisfied for all weighted graphs ðb; cþ over V. Our result reads as follows. We are not aware of an earlier result of this form in this context. Theorem 5. Let ðv; b; cþ be a weighted graph and m a measure on V of full support such that ðaþ holds. Then, for any p A ½1; yþ the operator L p is the restriction of ~L to DðL p Þ¼fu A l p ðv; mþ : ~Lu A l p ðv; mþg: Remark. The theory of Jacobi matrices already provides examples showing that without ðaþ the statement becomes false for p ¼ 2. This is discussed in Section 4.

88 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 11 The condition ðaþ does not imply that ~Lf belongs to l 2 ðv; mþ for all f A C c ðvþ. However, if this is the case, then ðaþ does imply essential selfadjointness. In this case, Q is the maximal form associated to the weighted graph ðb; cþ. More precisely, the following holds. Theorem 6. Let V be a set, m a measure on V with full support, ðb; cþ a weighted graph over V and Q the associated regular Dirichlet form. Assume ~LC c ðvþ L l 2 ðv; mþ. Then, DðLÞ contains C c ðvþ. If furthermore ðaþ holds, then the restriction of L to C c ðvþ is essentially selfadjoint and the domain of L is given by DðLÞ ¼fu A l 2 ðv; mþ : ~Lu A l 2 ðv; mþg and the associated form Q satisfies Q ¼ Q max, i.e., for all u A l 2 ðv; mþ. QðuÞ ¼ 1 2 P x; y A V bðx; yþ uðxþ uðyþ 2 þ P x A V cðxþuðxþ 2 Remark. (a) If inf mðxþ > 0 then both ðaþ and ~LC c ðvþ L l 2 ðv; mþ hold for any weighted graph ðb; cþ over V. In this case, we recover the corresponding results of [17], [26], [28] on essential selfadjointness, as these works assume m 1 1. (They also have additional restrictions on b but this is not relevant here). (b) The statement on the form being the maximal one seems to be new even in the context of [17], [26], [28]. (c) Essential selfadjointness fails in general if ðaþ does not hold as can be seen by examples (see Section 4 and the previous remark). 2. Dirichlet forms on graphs basic facts In this section we consider a countable set V together with a measure m of full support. Lemma 2.1. DðQÞ. Let Q be a regular Dirichlet form on ðv; mþ. Then, C c ðvþ is contained in Proof. Let x A V be arbitrary. Choose j A C c ðvþ with jðxþ ¼2 and jðyþ ¼0 for all y 3 x. AsC c ðvþ X DðQÞ is dense in C c ðvþ with respect to the supremum norm, there exists c A DðQÞ with cðxþ > 1 and jcðyþj < 1 for all y 3 x. AsQ is a Dirichlet form, DðQÞ is invariant under taking modulus and we can assume that c is non-negative. As Q is a Dirichlet form, also c ~ :¼ c51 belongs to DðQÞ. (Here, 5 denotes the minimum.) As DðQÞ is a vector space it contains c c ~ and this is a (nonzero) multiple of j by construction. As x A V was arbitrary, the statement follows. r Lemma 2.2. Let Q be a regular Dirichlet form on ðv; mþ. Then, there exists a weighted graph ðb; cþ over V such that the restriction of Q to C c ðvþ equals Q comp b; c.

89 12 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs Proof. By the previous lemma, C c ðvþ is contained in DðQÞ. Then, for any finite K L V, the restriction Q K of Q to C c ðkþ is a Dirichlet form as well. By standard results (see e.g. [1], Théorème 1), there exists then b K, c K with Q K ¼ Q comp b K ; c K. For K L K 0 and x; y A K it is not hard to see that b K ðx; yþ ¼b K 0ðx; yþ and c K ðxþ f c K 0ðxÞ. Thus, a simple limiting procedure gives the result. r Lemma 2.3. Let m be a measure on V of full support. Let ðb; cþ be a weighted graph over V. Then, Q max b; c; m is closed and Qcomp b; c is closable and its closure Q b; c; m is a restriction of Q max b; c; m. Proof. It su ces to show that Q max b; c; m is closed. Thus, it su ces to show lower semicontinuity of u 7! Q max b; c; mðu; uþ. This follows easily from Fatou s lemma. r Theorem 7. The regular Dirichlet forms on ðv; mþ are exactly given by the forms Q b; c; m with weighted graphs ðb; cþ over V. Proof. By the previous lemma and the discussion in Section 1, any Q b; c; m is a regular Dirichlet form. The converse follows from the previous lemmas. r The study of regular Dirichlet forms on ðv; mþ is based on first understanding their restrictions to finite sets. This is done next. Proposition 2.4. Let ðv; mþ be given and ðb; cþ a weighted graph over V. Let K H V be finite. Then, L ðdþ K is a bounded operator with L ðdþ K f ðxþ ¼ 1 P bðx; yþ f ðxþ f ðyþ P þ bðx; yþþcðxþ f ðxþ : mðxþ y A K y A VnK In particular, ~Li K f ðxþ ¼L ðdþ K f ðxþ for all x A K, where i K : l 2 ðk; m K Þ!l 2 ðv; mþ is the canonical embedding by extension by zero. Proof. Every linear operator on the finite dimensional l 2 ðk; m K Þ is bounded. Thus, we can directly read o the operator L ðdþ K from the form QðDÞ K given by QðDÞ K ðuþ :¼ Qði KuÞ. This gives the first claim. The last statement follows easily. r We now discuss two results on solutions of the associated di erence equation. These results will be rather useful for our further considerations. We start with a version of a minimum principle. Theorem 8 (Minimum principle). Let ðv; b; cþ be a weighted graph and m a measure on V of full support. Let U L V be given. Assume that the function u on V satisfies: ð~l þ aþu f 0 on U for some a > 0. The negative part uu :¼ uj U50 of the restriction of u to U attains its minimum on each connected component of U. u f 0 on VnU. Then, u 1 0 or u > 0 on each connected component of U. In particular, u f 0.

90 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 13 Proof. Without loss of generality we can assume U is connected. If u > 0 there is nothing left to show. It remains to consider the case that there exists x A U with uðxþ e 0. As the negative part of u on U attains its minimum, there exists then x m A U with uðx m Þ e 0 and uðx m Þ e uðyþ for all y A U. As uðyþ f 0 for y A U c, we obtain uðx m Þ uðyþe0 for all y A V. By the supersolution assumption we find 0 e P bðx m ; yþ uðx m Þ uðyþ þ cðx m Þuðx m Þþamðx m Þuðx m Þ e 0: As b and c are non-negative and a > 0, we find 0 ¼ uðx m Þ and uðyþ ¼uðx m Þ¼0 for all y with bðy; x m Þ > 0. As U is connected, iteration of this argument shows u 1 0onU. r The following lemma will be a key tool in our investigations. Note that its proof is rather simple due to the discreteness of the underlying space. Lemma 2.5 (Monotone convergence of solutions). Let a A R, f : V! R and u : V! R be given. Let ðu n Þ be a sequence of non-negative functions on V belonging to the set ~F given in (1) on which ~L is defined. Assume u n e u nþ1 for all n A N, and u n ðxþ!uðxþ and ð~l þ aþu n ðxþ! f ðxþ for all x A V. Then, u belongs to ~F as well and the equation ð~l þ aþu ¼ f holds. Proof. Without loss of generality we assume m 1 1. By assumption ð~l þ aþu n ðxþ ¼ P bðx; yþ u n ðxþ u n ðyþ þ cðxþþa u n ðxþ y A V P converges to f ðxþ for any x A V. As bðx; yþu n ðxþ converges increasingly to y A V uðxþ P P bðx; yþ < y, the assumptions on u n show that bðx; yþu n ðyþ must converge y A V y A V as well and in fact must converge to P bðx; yþuðyþ. From this we easily obtain the statement. r y A V We next discuss some fundamental properties of regular Dirichlet forms. These properties do not depend on the graph setting. They are true for general Dirichlet forms and can, for example, be found in the works [24], [25]. For the convenience of the reader we include short proofs based on the previous minimum principle. Proposition 2.6 (Domain monotonicity). Let ðv; b; cþ be a weighted graph and m a measure of full support. Let K 1 ; K 2 L V finite with K 1 L K 2 be given. Then, for any x A K 1, ðl ðdþ K1 þ aþ 1 f ðxþ e ðl ðdþ K 2 þ aþ 1 f ðxþ for all f A l 2 ðv; mþ with f f 0 and supp f L K 1. Proof. Consider f A l 2 ðv; mþ with f f 0 and supp f L K 1 and define u i :¼ðL ðdþ K i þ aþ 1 f ; i ¼ 1; 2. Extending u i by zero we can assume that u i are defined on the whole of V. Then, ð~l þ aþu i ¼ f on K i

91 14 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs for i ¼ 1; 2. Therefore, w :¼ u 2 u 1 satisfies w ¼ u 2 f 0onK1 c. The negative part of w attains its minimum on K 1 (as K 1 is finite). ð~l þ aþw ¼ f f ¼ 0onK 1. The minimum principle yields w f 0onV. r Regularity is crucial for the proof of the following result. Proposition 2.7 (Convergence of resolvents/semigroups). Let ðv; b; cþ be a weighted graph, m a measure on V with full support and Q the associated regular Dirichlet form. Let ðk n Þ be an increasing sequence of finite subsets of V with V ¼ S K n. Then, ðl ðdþ K n þ aþ 1 f!ðlþaþ 1 f, n! y for any f A l 2 ðk 1 ; m K1 Þ. (Here, ðl ðdþ K n þ aþ 1 f is extended by zero to all of V.) The corresponding statement also holds for the semigroups. Proof. By general principles (see e.g. [27], Satz 9.20b) it su ces to consider the resolvents. After decomposing f in positive and negative part, we can restrict attention to f f 0. Define u n :¼ðL ðdþ K n þ aþ 1 f. Then, u n f 0. Now, by standard characterization of resolvents (see e.g. [14], Section 1.4), u n is the unique minimizer of Q Kn ðuþþa u 1 a f 2 : By domain monotonicity, the sequence u n ðxþ is monotonously increasing for any x A V. Moreover, by standard results on Dirichlet forms (see e.g. [14], Theorem 1.4.1), we have u n e 1 a k f k y and by the spectral theorem ku nk e 1 a k f k. Thus, the sequence u n converges pointwise and in l 2 ðv; mþ towards a function u A l 2 ðv; mþ. Let now w A C c ðvþ be arbitrary. Assume without loss of generality that the support of w is contained in K 1. Then, QðwÞ ¼Q Kn ðwþ for all n A N. Closedness of Q, convergence of the ðu n Þ and the minimizing property of each u n then give QðuÞþa u 1 a f 2 e lim inf n!y ¼ lim inf n!y ¼ lim inf n!y e lim inf n!y Qðu nþþa u 1 a f Qðu nþþa u n 1 a f 2 2! Q K n ðu n Þþa u n 1 a f Q K n ðwþþa w 1 a f ¼ QðwÞþa w 1 a f 2 : 2! 2!

92 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 15 As w A C c ðvþ is arbitrary and Q is regular (!), this implies QðuÞþa u 1 a f for any v A DðQÞ. Thus, u is a minimizer of QðuÞþa u 1 a f 2 e QðvÞþa v 1 a f 2 : 2 By characterization of resolvents again, u must then be equal to ðl þ aþ 1 f. r We can use the previous result to connect the operator L to the formal operator ~L. To do so we need one further result. Lemma 2.8. Let ðv; mþ be given and ðb; cþ a weighted graph over V. Let p A ½1; yš be given. For any g A l p ðv; mþ, the function u :¼ ðl p þ aþ 1 g belongs to the set ~F given in (1) on which ~L is defined and solves ð~l þ aþu ¼ g. Proof. We first consider the case p ¼ 2. If su ces to consider the case g f 0. Choose an increasing sequence ðk n Þ of finite subsets of V with S K n ¼ V and let g n be the restriction of g to K n. Then, ðg n Þ converges monotonously increasing to g in l 2 ðv; mþ and consequently ðl þ aþ 1 g n converges monotonously increasing to u. Thus, by monotone convergence of solutions (Lemma 2.5), we can assume without loss of generality that g has compact support contained in K 1. By convergence of resolvents, u n :¼ðL ðdþ K n þ aþ 1 g then converges increasingly to u :¼ ðl þ aþ 1 g: Moreover, by Proposition 2.4, u n satisfies ð~l þ aþu n ¼ g on K n. Thus, the statement follows by monotone convergence of solutions. We now turn to general p A ½1; yš. Again, it su ces to consider the case g f 0. Choose an increasing sequence ðk n Þ of finite subsets of V with S K n ¼ V and let g n be the restriction of g to K n. Then, u n :¼ðL p þ aþ 1 g n converges to u. Moreover, as g n belongs to l 2 ðv; mþ consistency of the resolvents gives u n ¼ðL þ aþ 1 g n. Now, on the l 2 ðv; mþ level we can apply the considerations for p ¼ 2 to obtain ð~l þ aþu n ¼ð~L þ aþðl þ aþ 1 g n ¼ g n : Taking monotone limits now yields the statement. r After these preparations, we can now give the desired information on the generators. Theorem 9. Let ðv; b; cþ be a weighted graph and m a measure on V of full support. Let p A ½1; yš be given. Then, L p f ¼ ~Lf for any f A DðL p Þ. Proof. Let f A DðL p Þ be given. Then, g :¼ ðl p þ aþ f exists and belongs to l p ðv; mþ. By the previous lemma, f ¼ðL p þ aþ 1 g solves ð~l þ aþ f ¼ g ¼ðL p þ aþ f and we infer the statement. r

93 16 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs We also note the following by-product of our investigation (see [26], [28], [7] for this result for locally finite graphs). Corollary 2.9 (Positivity improving). Let ðv; b; cþ be a connected weighted graph and L the associated operator. Then, both the semigroup e tl, t f 0, and the resolvent ðl þ aþ 1, a > 0, are positivity improving (i.e., they map non-negative nontrivial l 2 -functions to strictly positive functions). Proof. By general principles it su ces to consider the resolvent. Let f A l 2 ðv; mþ with f f 0 be given and consider u :¼ ðl þ aþ 1 f. Then u f 0 as the resolvent of a Dirichlet form is positivity preserving. If u is not strictly positive, there exists an x with uðxþ ¼0. As u is non-negative, u attains its minimum in x. By Lemma 2.8, u satisfies ð~l þ aþu ¼ f f 0. We can therefore apply the minimum principle (with U ¼ V) to obtain that u 1 0. This implies f 1 0. r 3. Generators of the semigroups on l p and essential selfadjointness on l 2 In this section we will consider a symmetric weighted graph ðv; b; cþ and a measure m on V of full support. We will be concerned with explicitly determining the generators of the semigroups on l p and studying essential selfadjointness of the generator on l 2. Both issues will be tackled by proving uniqueness of solutions on the corresponding l p spaces. The results of this section are not needed to deal with stochastic completeness. Recall the geometric assumption introduced in the first section: (A) The equality P mðx n Þ¼y holds for any sequence ðx n Þ of elements of V such n A N that bðx n ; x nþ1 Þ > 0 for all n A N. The relevance of ðaþ comes from the following variant of the minimum principle: Proposition 3.1. Assume ðaþ. Let a > 0, p A ½1; yþ and u A l p ðv; mþ with ð~l þ aþu f 0 be given. Then, u f 0. Proof. Assume the contrary. Then, there exists an x 0 A V with uðx 0 Þ < 0. By 0 e ð~l þ aþuðx 0 Þ¼ 1 P mðx 0 Þ y A V bðx 0 ; yþ uðx 0 Þ uðyþ þ cðx 0Þ mðx 0 Þ uðx 0Þþauðx 0 Þ there must exist an x 1 connected to x 0 with uðx 1 Þ < uðx 0 Þ. Continuing in this way, we obtain a sequence ðx n Þ of connected points with uðx n Þ < uðx 0 Þ < 0. Combining this with ðaþ, we obtain a contradiction to u A l p ðv; mþ. r Let us note the following consequence of the previous minimum principle. Lemma 3.2 (Uniqueness of solutions on l p ). Assume ðaþ. Let a > 0, p A ½1; yþ and u A l p ðv; mþ with ð~l þ aþu ¼ 0 be given. Then, u 1 0.

94 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 17 Proof. u 1 0. r Both u and u satisfy the assumptions of the previous proposition. Thus, Remark. The situation for p ¼ y is substantially more complicated as can be seen by (part (ii) of) our first theorem. This lemma allows us to determine the generators whenever ðaþ holds. Proof of Theorem 5. Define ~D p :¼fu A l p ðv; mþ : ~Lu A l p ðv; mþg: By Theorem 9, we already know L p f ¼ ~Lf for any f A DðL p Þ. It remains to show ~D p L DðL p Þ: Let f A ~D p be given. Then, g :¼ ð~l þ aþ f belongs to l p ðv; mþ. Thus, u :¼ ðl p þ aþ 1 g belongs to DðL p Þ. Now, as shown above, see Lemma 2.8, u solves ð~l þ aþu ¼ g. Moreover, f also solves this equation. Thus, by the uniqueness of solutions given in Lemma 3.2, we infer f ¼ u and f belongs to DðL p Þ. This finishes the proof. r We now turn to a study of essential selfadjointness on C c ðvþ. Clearly, the question of essential selfadjointness on C c ðvþ only makes sense if ~LC c ðvþ L l 2 ðv; mþ. In this context, we have the following result: Proposition 3.3. Let ðv; mþ be given and ðb; cþ a weighted graph over V. Then, the following assertions are equivalent: (i) ~LC c ðvþ L l 2 ðv; mþ. (ii) For any x A V, the function V!½0; yþ, y 7! bðx; yþ=mðyþ, belongs to l 2 ðv; mþ. In this case, any u A l 2 ðv; mþ belongs to the set ~F of(1) on which ~L is defined and the three sums P P uðxþ ~LvðxÞmðxÞ; ~LuðxÞvðxÞmðxÞ and 1 2 P x; y A V x A V x A V bðx; yþ uðxþ uðyþ vðxþ vðyþ þ P x A V converge absolutely and agree for all u A l 2 ðv; mþ and v A C c ðvþ. cðxþuðxþvðxþ Proof. Without loss of generality we assume c 1 0. For any x A V define d x : V! R by d x ðyþ ¼1ifx¼ y and d x ðyþ ¼0ifx3y. Obviously, (i) is equivalent to ~Ld x A l 2 ðv; mþ for all x A V. This latter condition can easily be seen to be equivalent to (ii). This shows the stated equivalence.

95 18 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs give Let u A l 2 ðv; mþ be given. Then, for any x A V, Cauchy Schwarz inequality and (ii) ð*þ P y A V P bðx; yþ 2 1=2 P 1=2 jbðx; yþuðyþj e uðyþ mðyþ 2 < y: y A V mðyþ y A V Thus, u belongs to ~F. To show the statement on the sums, it su ces to consider u A l 2 ðv; mþ and v ¼ d z, for z A V arbitrary. In this case, the desired statements can easily be reduced to the question of absolute convergence of P x; y A V bðx; yþuðxþd z ðxþ and P x; y A V bðx; yþuðxþd z ðyþ: This absolute convergence in turn is shown in ( * ). r Proof of Theorem 6. As ~LC c ðvþ L l 2 ðv; mþ, we can define the minimal operator L min to be the restriction of ~L to DðL min Þ :¼ C c ðvþ and the maximal operator L max to be the restriction of ~L to The previous proposition gives for all u; v A C c ðvþ. This extends to give DðL max Þ :¼ fu A l 2 ðv; mþ : ~Lu A l 2 ðv; mþg: hu; L min vi ¼ Q comp b; c ðu; vþ hu; L min vi ¼ Q b; c; m ðu; vþ for all u A DðQÞ and v A C c ðvþ. Thus, L min is a restriction of L in this case. Moreover, the previous proposition gives also hu; L min vi ¼ P x A V ~LuðxÞvðxÞmðxÞ for all v A C c ðvþ and u A l 2 ðv; mþ. Thus, L min ¼ L max: Thus, essential selfadjointness of L min is equivalent to selfadjointness of L max. This in turn is equivalent to L ¼ L max (as we have L L L max by Theorem 5). As ðaþ and Theorem 5 yield DðLÞ ¼fu A l 2 ðv; mþ : ~Lu A l 2 ðv; mþg, we infer L ¼ L max and essential selfadjointness of the restriction of L to C c ðvþ (¼ L min ) follows.

96 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 19 It remains to show the statement on the form. Let Q max be the maximal form, i.e., Q max ðuþ ¼ 1 2 P x; y A V bðx; yþ uðxþ uðyþ 2 þ P x A V cðxþuðxþ 2 for all u A l 2 ðv; mþ and L Q max previous proposition shows the associated operator. Then, another application of the P x A V ~LuðxÞvðxÞmðxÞ ¼ 1 2 P x; y A V bðx; yþ uðxþ uðyþ vðxþ vðyþ þ P x A V cðxþuðxþvðxþ ¼ Q max ðu; vþ ¼ hl Q maxu; vi associ- for all u A DðL Q maxþ and v A C c ðvþ. This gives that the self-adjoint operator L Q max ated to Q max satisfies L Q maxu ¼ ~Lu ¼ Lu for all u A DðL Q maxþ. Thus, L Q max L L: As L is selfadjoint, we infer L Q max ¼ L and the statement on the form follows. r 4. Some counterexamples In this section, we first discuss an example showing that without condition ðaþ Theorem 5 and Theorem 6 fail in general. We then present an example of a non-regular Dirichlet form on a weighted graph. Note that the choice of the measure plays a crucial role here. Example for failure of Theorem 5 and 6 without assumption (A). first) every point of Z have measure 1. Consider the bounded operator Let V 1 Z. Let (at D : l 2 ðzþ!l 2 ðzþ; ðdjþðxþ ¼ jðx 1Þþ2jðxÞ jðx þ 1Þ: It corresponds to the Dirichlet form Q b; 0; 1 with bðx; yþ ¼1 whenever jx yj ¼1. A direct calculation shows that the function u : V! R; uðxþ :¼ e lx is a positive solution to the equation ð ~ D þ aþu ¼ 0 for a ¼ e l þ e l 2. Obviously, we have a b 0 for all real l. Now let w A l 1 ðzþ, w > 0 and define the measure m : V!ð0; yþ; mðxþ :¼ min 1; wðxþ u 2 ðxþ

97 20 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs and the killing term c : V!½0; yþ; cðxþ :¼ max 0; u2 ðxþ wðxþ 1 amðxþ: By construction, u then belongs to l 2 ðz; mþ and a m þ c m þ a 1 0: Let ~L be defined by ~LvðxÞ :¼ 1 P mðxþ y A V bðx; yþ vðxþ vðyþ þ cðxþ mðxþ vðxþ; i.e., in a formal sense ~L ¼ 1 m ð~ D þ cþ. Then, the restriction L max of ~L to has the eigenvalue a < 0 since fv A l 2 ðz; mþ : ~Lv A l 2 ðz; mþg ðl max þ aþuðxþ ¼ a m þ c m þ a uðxþ ¼0: Consider now the operator L associated with the Dirichlet form Q b; c; m on l 2 ðz; mþ. Of course, L is a positive operator and therefore can not have a negative eigenvalue. Moreover, by the results of the previous section, this operator is a restriction of ~L. This implies that u can not belong to DðLÞ and therefore DðLÞ 3 DðL max Þ. Thus, the domain of definition DðLÞ is not given by Theorem 5. In this case, the restriction of ~L to C c ðvþ is not essentially self-adjoint (as the proof of Theorem 6 showed that otherwise L ¼ L max ). Example of a non-regular Dirichlet form on V. We consider connected graphs ðv; b; cþ with c 1 0 and bðx; yþ A f0; 1g for all x; y A V. As discussed by Dodziuk Kendall [10] (see [8], [18] as well) in the context of isoperimetric inequalities, any such graph with positive Cheeger constant a > 0 has the property that 1 2 P x; y A V for all j A C c ðvþ, where dðxþ ¼ P bðx; yþ jðxþ jðyþ 2 f a 2 y A V 2 P x A V dðxþjðxþ 2 bðx; yþ. Let now such a graph be given. Fix an arbitrary x 0 A V. Choose a measure m with support V and mðvþ ¼1. Thus, the constant function 1 belongs to l 2 ðv; mþ. Define the form Q by QðuÞ :¼ 1 2 P x; y A V bðx; yþ uðxþ uðyþ 2

98 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 21 for all u A l 2 ðv; mþ for which QðuÞ is finite. Obviously, Q is a Dirichlet form and the constant function 1 satisfies Qð1Þ ¼0. Let now j n be any sequence in C c ðvþ converging to 1 in l 2 ðv; mþ. Then, j n ðx 0 Þ converges to 1. In particular, Qðj n Þ f a2 2 dðx 0Þj n ðx 0 Þ 2! a2 2 dðx 0Þ > 0; n! y: Thus, Qðj n Þ does not converge to 0 ¼ Qð1Þ. Hence, Q is not regular. 5. The heat equation on l T In this section we consider a weighted graph ðb; cþ over the measure space ðv; mþ with associated formal operator ~L. A function N : ½0; yþv! R is called a solution of the heat equation if for each x A V the function t 7! N t ðxþ is continuous on ½0; yþ and di erentiable on ð0; yþ and for each t > 0 the function N t belongs to the domain of ~L and the equality d dt N tðxþ ¼ ~LN t ðxþ holds for all t > 0 and x A V. For a bounded solution N continuity of ½0; yþ!r, t 7! ~LN t ðxþ, can easily be seen (for each fixed x A V). For such N validity of d dt N tðxþ ¼ ~LN t ðxþ for t > 0 then extends automatically to t ¼ 0, i.e., t 7! N t ðxþ is di erentiable on ½0; yþ and d dt N tðxþ ¼ ~LN t ðxþ holds for any t f 0. The following theorem is essentially a standard result in the theory of semigroups. In the situation of special graphs it has been shown in [26], [28]. For completeness reason we give a proof in our situation as well. Theorem 10. Let L be a self-adjoint restriction of ~L, which is the generator of a Dirichlet form on l 2 ðv; mþ. Let v be a bounded function on V and define N : ½0; yþv! R by N t ðxþ :¼ e tl vðxþ. Then, the function NðxÞ : ½0; yþ!r, t 7! N t ðxþ, is di erentiable and satisfies for all x A V and t f 0. d dt N tðxþ ¼ ~LN t ðxþ Proof. As v is bounded, continuity of t 7! N t ðxþ follows from general principles on weak l 1 -l y continuity of the semigroup on l y ðv; mþ, see e.g. [6]. It remains to show di erentiability and the validity of the equation. As discussed already, it su ces to consider t > 0. After decomposing v into positive and negative part, we can assume without loss of generality that v is non-negative.

99 22 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs Let ðk n Þ be a sequence of finite increasing subsets of V with S K n ¼ V. Let v n be the function on V which agrees with v on K n and equals zero elsewhere. Thus, v n belongs to l 2 ðv; mþ and we can consider e tl v n for any n A N. For each fixed x A V the function t 7! e tl v n ðxþ converges monotonously to t 7! N t ðxþ (by definition of the semigroup on l y ). As t 7! N t ðxþ is continuous, this convergence is even uniform on compact subintervals of ð0; yþ. Moreover, standard l 2 -theory shows that N n ¼ e tl v n satisfies d dt N nðxþ ¼ LN n ðxþ for all t > 0 and x A V. By assumption L is a restriction of ~L and we infer d dt N nðxþ ¼ 1 mðxþ P y A V bðx; yþ N n ðxþ N n ðyþ cðxþ mðxþ N nðxþ for all x A V and t > 0. This equality together with the uniform convergence on compact intervals in ð0; yþ and the summability of the bðx; yþ in y gives uniform convergence of the d dt N nðxþ on compact intervals. Hence, t! N t ðxþ is di erentiable on ð0; yþ and satisfies the desired equation. r Lemma 5.1. Let N be a bounded solution of d dt N þ ~LN ¼ 0, N Then, for arbitrary a > 0, the function v :¼ Ðy e ta N t dt solves ð~l þ aþv ¼ 0. 0 Proof. This follows by a short calculation: By boundedness of N and P y we can interchange two limits to obtain bðx; yþ < y, Ð ~LvðxÞ T ¼ lim e ta ~LN t ðxþ dt: T!y 0 Using that N solves the heat equation and partial integration we find ~LvðxÞ ¼ lim T!y Ð T 0 e ta d dt N tðxþ dt ¼ lim T!y e ta N t ðxþj T 0 ÐT 0 ae ta N t ðxþ dt ¼ a Ðy 0 e ta N t ðxþ dt ¼ avðxþ: Here, we used boundedness of N and N 0 ¼ 0 to get rid of the boundary terms after the partial integration. r 6. Extended semigroups and resolvents We are now going to extend the resolvents/semigroups to a larger class of functions. To do so, we note that for a function f on V with f f 0 the functions g A C c ðvþ

100 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 23 with 0 e g e f form a net with respect to the natural ordering g h whenever g e h. Limits along this net will be denoted by lim. As the resolvents and semigroups are positivity g f preserving, for f f 0, a > 0, t > 0, we can define the functions ðl þ aþ 1 f : V!½0; yš and e tl f : V!½0; yš by and ðl þ aþ 1 f ðxþ :¼ lim g f ðl þ aþ 1 gðxþ e tl f ðxþ :¼ lim g f e tl gðxþ: In fact, as we are in a discrete setting, the operators have kernels, i.e., for any t f 0 there exists a unique function e tl : V V!½0; yþ with e tl f ðxþ ¼ P y A V e tl ðx; yþ f ðyþ for any f f 0 (and similarly for the resolvent). It is not hard to see that for functions in l y ðvþ, these definitions are consistent with our earlier definitions. Theorem 11 (Properties of extended resolvents and semigroups). Let f be a non-negative function on V. Let a > 0 be given. (a) Let K n be an increasing sequence of finite subsets of V with S K n ¼ V. Let f n be the restriction of f to K n, and u n :¼ðL ðdþ K n þ aþ 1 f n. Then, u n converges pointwise monotonously to ðl þ aþ 1 f. (b) The following statement are equivalent: (i) There exists a non-negative l : V!½0; yþ with ð~l þ aþl f f. (ii) ðl þ aþ 1 f ðxþ is finite for any x A V. In this case u :¼ ðl þ aþ 1 f is the smallest non-negative function l with ð~l þ aþl f f and it satisfies ð~l þ aþu ¼ f. (c) For all x A V ðl þ aþ 1 f ðxþ ¼ Ðy 0 e ta e tl f ðxþ dt: Remark. Note that the functions in (a) and (c) are allowed to take the value y. Statement (a) is an extension of Proposition 2.7 to non-negative functions. Proof. Throughout the proof we let u denote the function ðl þ aþ 1 f.

101 24 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs (a) Let x A V be given. By domain monotonicity u n ðxþ ¼ðL ðdþ K n þ aþ 1 f n ðxþ is increasing. Moreover, again by domain monotonicity and f n e f we have u n ðxþ ¼ðL ðdþ K n þ aþ 1 f n ðxþ e ðl þ aþ 1 f n ðxþ e ðl þ aþ 1 f ðxþ ¼uðxÞ for all n. It remains to show the converse inequality. We consider two cases. Case 1. uðxþ < y. Let e > 0 be given. By definition of the extended resolvents there exists then g A C c ðvþ with 0 e g e f and uðxþ e e ðl þ aþ 1 gðxþ: As g has compact support, we can assume without loss of generality that the support of g is contained in K n for all n. By convergence of resolvents, we conclude ðl þ aþ 1 gðxþ e e ðl ðdþ K n for all su ciently large n. Thus, for such n we find uðxþ 2e e ðl ðdþ K n þ aþ 1 gðxþ þ aþ 1 gðxþ: By g e f and supp g L K 1, we have g e f n for all n. Thus, the last inequality gives uðxþ 2e e ðl ðdþ K n This finishes the considerations for this case. þ aþ 1 f n ðxþ ¼u n ðxþ: Case 2. uðxþ ¼y. Let k > 0 be arbitrary. By definition of the extended resolvents there exists then g A C c ðvþ with 0 e g e f and Now, we can continue as in Case 1 to obtain k e e ðl ðdþ K n k e ðl þ aþ 1 gðxþ: þ aþ 1 f n ðxþ ¼u n ðxþ for all su ciently large n. As k > 0 is arbitrary the statement follows. (b) We first show (ii) ) (i): Recall that u ¼ðL þ aþ 1 f and consider g A C c ðvþ with 0 e g e f. Then, by Lemma 2.8, u g :¼ðL þ aþ 1 g solves ð~l þ aþu g ¼ g: Taking monotone limits on both sides and using the finiteness assumption (ii), we obtain This shows (i) (with l ¼ u). ð~l þ aþu ¼ f :

102 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 25 We next show (i) ) (ii): Let l f 0 satisfy ð~l þ aþl f f. Let ðk n Þ be an increasing sequence of finite subsets of V as in (a) and let f n be the restriction of f to K n. Extend u n :¼ðL ðdþ K n þ aþ 1 f n by zero to all of V. Then, w n :¼ l u n satisfies: w n ¼ l f 0onK c n. The negative part of w n attains its minimum on K n (as K n is finite). ð~l þ aþw n ¼ð~L þ aþl ð~l þ aþu n f f f ¼ 0onK n. The minimum principle, Theorem 8, then gives w n ¼ l u n f 0: As n is arbitrary and u n converges to u by part (a), we find that u e l is finite. This finishes the proof of the equivalence statement of (b). The last statements of (b) have already been shown along the proofs of (i) ) (ii) and (ii) ) (i). (c) For g A C c ðvþ with 0 e g e f the equation ðl þ aþ 1 g ¼ Ðy 0 e ta e tl gdt holds by standard theory on semigroups. Now, (c) follows by taking monotone limits on both sides. r There is a special function v to which our considerations can be applied: Proposition 6.1. For any a > 0 we have the estimate 0 e ðl þ aþ 1 a1 þ c e 1: m Remark. As a1 þ c=m f 0, we have 0 e ðl þ aþ 1 ða1 þ c=mþ. Moreover, we obvi- Proof. ously have Let us stress that c=m is not assumed to be bounded. ð~l þ aþ1 ¼ a1 þ c m : Thus, (b) of the previous theorem shows ðl þ aþ 1 ða1 þ c=mþ e 1. r We will also need the following consequence of the proposition. Proposition 6.2. Let ðv; b; cþ be a weighted graph and define S : V!½0; yš by SðxÞ :¼ Ðy e sl c ðxþ ds: 0 m Then, S satisfies 0 e S e 1 and ~LS ¼ c=m.

103 26 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs Proof. For g A C c ðvþ with 0 e g e c=m and a > 0, we define S g; a by S g; a :¼ Ðy 0 e as e sl gðxþ ds and S g by S g ¼ lim a!0 S g; a. Then, S g; a ¼ðL þ aþ 1 g; i:e:; ð~l þ aþs g; a ¼ g: By g e a1 þ c=m for any a > 0 and Proposition 6.1, we have S g; a ¼ðLþaÞ 1 gðxþ e ðl þ aþ 1 a1 þ c ðxþ e 1: m As S g is the monotone limit of the S g; a, this shows that S g is bounded by 1. Moreover, using the uniform bound on the S g; a and taking the limit a! 0in we find ð~l þ aþs g; a ¼ g; ~LS g ¼ g f 0: As S ¼ lim gc=m S g and the S g are uniformly bounded, we obtain the statement. r Lemma 6.3. Let u f 0 be given. Then, the following assertions are equivalent: (i) e tl u e u for all t > 0. (ii) ðl þ aþ 1 u e 1 u for all a > 0. a Any u f 0 with ~Lu f 0 satisfies these equivalent conditions. Proof. The implication (i) ) (ii) follows easily from Theorem 11 (c) giving ðl þ aþ 1 ¼ Ðy e ta e tl dt. Similarly, the implication (ii) ) (i) follows by a limiting 0 argument from the standard e tl f ¼ lim n!y t n L þ n! n f t for f A l 2 ðv; mþ. As for the last statement, we note that ~Lu f 0 implies 1 a ð ~L þ aþu f u: By (b) of Theorem 11 the desired statement (ii) follows. r

104 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs Characterization of stochastic completeness In this section, we can finally characterize stochastic completeness. We begin by introducing the crucial quantity in our studies. Lemma 7.1. Let ðv; b; cþ be a weighted graph and m a measure on V with full support. Then, the function M : ½0; yþv!½0; yš defined by M t ðxþ :¼ e tl 1ðxÞþ Ðt e sl c ðxþ ds m satisfies 0 e M s e M t e 1 for all s f t f 0 and, for each x A V, the map t 7! M t ðxþ is di erentiable and satisfies d dt M tðxþþ~lm t ðxþ ¼cðxÞ=mðxÞ. 0 Remark. We can give an interpretation of M in terms of a di usion process on V as follows. For x A V, let d x be the characteristic function of fxg. A di usion on V starting 1 in x with normalized measure is then given by mðxþ d x at time t ¼ 0. It will yield to the amount of heat e tl d x mðxþ ; 1 ¼ d x mðxþ ; e tl 1 ¼ P y A V e tl ðx; yþ ¼e tl 1ðxÞ within V at the time t. Moreover, at each time s the rate of heat killed at y by the killing term c is given by e sl ðx; yþcðyþ=mðyþ. The total amount of heat killed at y until the time t is then given by Ðt e sl ðx; yþcðyþ=mðyþ ds. The total amount of heat killed at all vertices 0 by c till the time t is accordingly given by P Ð t y A V 0 e sl ðx; yþ cðyþ mðyþ ds ¼ Ðt P 0 y A V e sl ðx; yþ cðyþ mðyþ Ðt ds ¼ e sl c m 0 ðxþ ds: This means that M measures the amount of heat at time t which has not been transferred to the boundary of V. Proof. By definition we have M f 0. By Proposition 6.2, SðxÞ ¼ Ðy ðe sl c=mþðxþ ds is finite and we can therefore calculate 0 Ð t 0 e sl c m ðxþ ds ¼ SðxÞ Ðy t e sl c ðxþ ds ¼ SðxÞ e tl SðxÞ; m where the last statement follows by taking monotone limits along the net of g A C c ðvþ with 0 e g e c=m. Thus, M t ¼ e tl 1 þ S e tl S ¼ S þ e tl ð1 SÞ:

105 28 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs From this equality the desired statements follow easily: By Proposition 6.2, we have 1 S f 0 and ~Lð1 SÞ ¼~L1 ~LS ¼ c=m c=m ¼ 0. Lemma 6.3 then yields e sl ð1 SÞ e e tl ð1 SÞ e 1 S for all s f t f 0. Plugging this into the formula for M t gives, for all 0 e t e s, 0 e M s e M t e 1: Moreover, as S and the constant function 1 are bounded, we can apply Theorem 10 to M t ¼ e tl ð1 SÞþS to infer that t 7! MðxÞ is di erentiable with d dt M tðxþ ¼ ~Le tl 1ðxÞþ~Le tl SðxÞ ¼ ~LM t ðxþþ~lsðxþ ¼ ~LM t ðxþþ cðxþ mðxþ ; where we used ~LS ¼ c=m from Proposition 6.2. r We now show that integration over M yields a resolvent. Lemma 7.2. ðl þ aþ 1 ða1 þ c=mþðxþ ¼ Ðy 0 ae ta M t ðxþ ds: Proof. As shown in (c) of Theorem 11 we have ðl þ aþ 1 a1 þ c ðxþ ¼ Ðy ae ta e tl 1 þ c ðxþ dt: m 0 am Thus, it su ces to show that Ðy e ta 0 e tl c m ðxþ dt ¼ Ðy 0 Ð ae ta t e sl c m 0 ðxþ ds dt: This follows by partial integration applied to each (non-negative) term of the sum e tl c ðxþ ¼ P e tl ðx; yþ cðyþ m y A V mðyþ : This finishes the proof. r Remark. Let us stress that the care taken with monotone convergence in the above arguments is quite necessary. For example one might think that 1 ¼ðLþaÞ 1 ð~l þ aþ1. Combined with the previous lemma, this would lead to 1 ¼ðL þ aþ 1 ða1 þ c=mþ ¼ Ðy 0 ae ta M t dt: However, the phenomenon we study is exactly that the integral can be strictly smaller than 1!

106 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 29 After these preparations we now prove our first main result. Recall that we defined w ¼ Ðy 0 ae ta ð1 M t Þ dt: Proof of Theorem 1. As Ðy 0 ae ta dt ¼ 1, the previous lemma gives w ¼ 1 ðl þ aþ 1 ða1 þ c=mþ: Thus, w solves ð~l þ aþw ¼ 0. Moreover, the minimality properties of the extended resolvent (see Theorem 11(b)) yield the maximality property of w. More precisely, let l be any non-negative function bounded by 1 with ð~l þ aþl e 0. Then, 1 l is non-negative and satisfies ð~l þ aþð1 lþ ¼a1 þ c m ð ~L þ aþl f a1 þ c m : The minimality property of 1 w ¼ðL þ aþ 1 ða1 þ c=mþ then gives 1 w e 1 l, and the desired inequality l e w follows. It remains to show the equivalence statements. (v) ) (iv): This is clear as 0 e M t e 1 and M is continuous. (iv) ) (iii): This is clear from the properties of w shown above. (iii) ) (ii): This is clear. (ii) ) (i): Let l þ be the positive part of l, i.e., l þ ðxþ ¼lðxÞ if lðxþ > 0 and l þ ðxþ ¼0 otherwise. If l þ is trivial, the function l is a nontrivial, non-negative bounded solution and (i) follows. Otherwise a direct calculation shows that l þ is a nontrivial subsolution. Obviously, l þ is non-negative and bounded. (i) ) (v): If there exists a nontrivial non-negative subsolution, then w as the largest subsolution must be nontrivial. Hence, there must exist t > 0 and x A V with M t ðxþ < 1. (v) ) (vi): Lemma 7.1 gives that N :¼ 1 M satisfies N 0 ¼ 0 and d dt N þ ~LN ¼ 0. This gives the desired implication. (vi) ) (i): This is a direct consequence of Lemma 5.1. r 8. Stochastically complete graphs with incomplete subgraphs In this section we prove Theorem 2 and Theorem 3. For Theorem 3 the basic idea is to attach graphs satisfying ðsc y Þ to each vertex of a graph (with ðsi y Þ) such that the resulting graph will satisfy ðsc y Þ. As adding a potential to a graph can be interpreted as adding edges to infinity, the proof of Theorem 2 can be seen as a variant of the proof of Theorem 3.

107 30 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs The graphs we attach will be given as follows. Let ðn; b N ; 0Þ the graph with vertex set N ¼f0; 1; 2;...g, b N ðx; yþ ¼1ifjx yj ¼1 and b N ðx; yþ ¼0 otherwise and c 1 0. Moreover, let the measure m on N be constant m 1 1. The next lemma shows that when u solves ð~l N þ aþuðxþ ¼0 for some a > 0 and all x A Nnf0g then it is only bounded if it is exponentially decreasing. Lemma 8.1. Let ðn; b N ; 0Þ be as above and m 1 1. Let a > 0 be given. Let u be a positive function on N with ð~l N þ aþuðxþ ¼0 for all x f 1. If for some x f 1, uðxþ f 2 uðx 1Þ; 2 þ a then u increases exponentially. Proof. Let u be a positive solution. If 1 þ a uðxþ f uðx 1Þ for some x f 1, we 2 get by the equation ð~l N þ aþuðxþ ¼0: 0 ¼ 1 þ a uðxþ uðxþ1þþ 1 þ a uðxþ uðx 1Þ 2 2 f 1 þ a uðxþ uðxþ1þ: 2 This implies uðx þ 1Þ f 1 þ a uðxþ and, in particular, 1 þ a uðx þ 1Þ f uðxþ. By 2 2 induction we then get for y f x: y x uðxþ uðyþ f 1 þ a 2 which gives the statement. r Proof of Theorem 3. Let ðw; b W ; c W Þ be a weighted graph and m a measure of full support on W. IfðSC y Þ holds we are done, so we assume the contrary. We will construct a weighted graph ðv; b; cþ satisfying ðsc y Þ such that W L V and bj WW ¼ b W. Define deg bw ðxþ ¼ 1 P bðx; yþ: mðxþ y A W Let n : W!ð0; yþ be a function which satisfies nðxþ deg bw ðxþmðxþ A N and P y j¼1 nðx j Þ¼y for any sequence ðx j Þ in W. (For example we can set nðxþ ¼½deg bw m þ 1Š=deg bw m, where ½xŠ denotes the smallest integer not exceeding x.) To each vertex x A W, we attach nðxþ deg bw ðxþmðxþ copies of the weighted graph ðn; b N ; 0Þ defined in the beginning of the section. We do this by identifying x A W with

108 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 31 the vertices 0 in the associated copies of N. We denote the resulting graph by V and define b on V V by letting 8 < b W ðx; yþ : x; y A W; bðx; yþ ¼ b N ðx; yþ : x; y in the same copy of N; : 0 : otherwise: Moreover, we extend c and m to V by letting c 1 0 and m 1 1onVnW and denote ~L ¼ ~L V. We will show that for all a > 0 every non-negative nontrivial function u on V, which satisfies ð~l þ aþu ¼ 0, is unbounded. Without loss of generality, we can assume that the graph is connected. Then, any non-negative nontrivial solution u of ð~l þ aþu ¼ 0 must be positive by the minimum principle. Let u be such a positive solution of ð~l þ aþu ¼ 0 and assume it is bounded. Fix x 0 A W and a sequence ðr r Þ in R with ð2 þ aþ=2 > r r > 1 and P ðr r 1Þ < y. By induction we can now define for each r A N an x r A V such that bðx r ; x r 1 Þ > 0 and r r uðx rþ1 Þ f uðyþ. Since we assumed u bounded, Lemma 8.1 gives sup y A V; bðx r ; yþ>0 uðyþ < 2uðx r Þ=ð2 þ aþ for each vertex y in a copy of N which is adjacent to x r.ifx rþ1 was in a copy of N, then this would imply that u has a maximum in x r which leads to a contradiction to ð~l þ aþu ¼ 0. Thus all x r belong to W. The equation ð~l þ aþuðx r Þ¼0 now gives 0 ¼ 1 P mðx r Þ y A V bðx r ; yþ uðx r Þ uðyþ þ cðx rþ mðx r Þ uðx rþþauðx r Þ f deg bw ðx r Þuðx r Þþ 1 P bðx r ; yþ uðx r Þ uðyþ P mðx r Þ y A VnW f 1 þ anðx rþ deg 2 þ a bw ðx r Þuðx r Þ r r deg bw ðx r Þuðx rþ1 Þ: y A W bðx r ; yþuðyþ In the second inequality, we used a; cðx r Þ; uðx r Þ f 0. In the third inequality, we estimated the sum over y A VnW by the inequality uðyþ < 2=ð2 þ aþuðx r Þ of Lemma 8.1 and the sum over y A W by the choice of x rþ1. We get by direct calculation and iteration uðx rþ1 Þ f 1 anðx r Þ Q r r 2 þ a þ 1 r uðx r Þ f j¼1 r j 1 Q r j¼1! anðx j Þ 2 þ a þ 1 uðx 0 Þ: Letting r tend to infinity the right-hand side diverges if and only if n is chosen such that P y j¼1 nðx j Þ is divergent. (Notice that the infinite product over ð1=r j Þ is greater than zero since we assumed that ðr j 1Þ is summable.) Thus, by our choice of n, we arrive at the contradiction that u is unbounded. By Theorem 1, this construction shows that for every ðw; b W ; c W Þ there is a weighted graph ðv; b; cþ which is ðsc y Þ and ðw; b W ; c W Þ is a subgraph of ðv; b; cþ. r Remark. An alternative construction is to add single vertices instead of copies of N. For the resulting graph and a function u satisfying ð~l þ aþu ¼ 0 the value of u on an added vertex y adjacent to the vertex x in the original graph is then determined by

109 32 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs ð1 þ aþuðyþ ¼uðxÞ. The rest of the proof can now be carried out in a similar manner. We chose to do the construction above to avoid the impression that the ðsc y Þ is the result of adding some type of boundary to the graph. We finish this section by proving Theorem 2. Proof of Theorem 2. Set bðxþ :¼ P y A V bðx; yþ. Choose c 0 : V!½0; yþ such that for any sequence ðx j Þ in V satisfying bðx j ; x jþ1 Þ > 0 for all j A N, we have P y j¼1 cðx j Þþc 0 ðx j Þ bðx j Þ ¼ y: (For example one may choose c 0 ðxþ ¼bðxÞ for x A V.) We now follow a similar reasoning as in the proof of Theorem 3: We consider a nontrivial non-negative solution u of ð~l b; cþc 0 ; m þ aþu ¼ 0, a > 0 and choose inductively for each r A N an x r A V and 2 f r r > 1 with uðx 0 Þ > 0, bðx r ; x rþ1 Þ > 0 and r r uðx rþ1 Þ f sup uðyþ for all r A N. y:bðx r ; yþ>0 Then, a direct calculation gives uðx rþ1 Þ f 1 1 þ cðx rþþc 0 ðx r Þ uðx r Þ and unboundedness r r bðxþ of u follows (whenever r r converges to 1 su ciently fast). Hence, by Theorem 1 the graph ðv; b; c þ c 0 Þ satisfies ðsc y Þ. r 9. An incompleteness criterion In this section we prove Theorem 4, which is a counterpart to Theorem 3. As shown there a subgraph with ðsi y Þ is well compatible with the whole graph satisfying ðsc y Þ. Theorem 4 shows under which additional condition ðsi y Þ of a subgraph implies ðsi y Þ for the whole graph. This condition is about how heavily the incomplete subgraph is connected with the rest of the graph. Not having control over the amount of connections leads possibly to ðsc y Þ as we have seen in Theorem 3. For a subset W of a weighted graph ðv; b; cþ we define the outer boundary qw of W in V by qw ¼fx A VnW : by A W; bðx; yþ > 0g: Note that the outer boundary of W is a subset of VnW. We will be concerned with decompositions of the whole set V into two sets W and W 0 :¼ VnW. In this case, there are two outer boundaries. Our intention is to extend positive bounded functions u on W with ð~l ðdþ W þ aþu e 0 to positive bounded functions v on the whole space satisfying ð~l þ aþv e 0. To do so, we will have to take particular care at what happens on the two boundaries. Lemma 9.1. Let ðv; b; cþ be a connected weighted graph. Let W L V be non-empty. Then, any connected component of W 0 ¼ VnW contains a point x A qw. Proof. Choose x A W arbitrarily. By assumption, any y A W 0 is connected to x by a path in V, i.e., there exist x 0 ; x 1 ;...; x n A V with bðx i ; x iþ1 Þ > 0 and x 0 ¼ x, x n ¼ y. Let

110 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 33 m A f0;...; ng be the largest number with x m A W. Then, x mþ1 belongs to both the boundary of W and the connected component of y. r Lemma 9.2. Let ðv; b; cþ be a weighted graph and m a measure of full support. Let U L V and j be a non-negative function in l 2 ðu; mþ. Then, ðl ðdþ U þ aþ 1 j is non-negative on U and positive on the connected components of any x A U with jðxþ > 0. Proof. The operator L ðdþ U is associated to the weighted graph ðu; bðdþ U ; cðdþ U Þ. Hence, Corollary 2.9 gives the statement. r Lemma 9.3. Let ðv; b; cþ be a weighted graph and m a measure of full support. Let U L V be given. Let v A ~F and denote the restriction of v to U by u and the restriction of v to VnU byu 0. Then, for any x A U, ð~l þ aþvðxþ ¼ð~L ðdþ U þ aþuðxþ 1 mðxþ Proof. This follows by direct calculation. r P y A VnU bðx; yþu 0 ðyþ: Proof of Theorem 4. Let W L V be given such that for every a > 0 there is a bounded non-negative nontrivial function u on W satisfying ð~l ðdþ W þ aþu e 0: By Theorem 1, it su ces to show that any such u can be extended to a non-negative and bounded function v on V such that ð~l þ aþv e 0: To do so, we proceed as follows: Set W 0 ¼ VnW. Define c : W 0! R; cðxþ ¼ 1 P bðx; yþuðyþ: mðxþ Thus, c vanishes on W 0 nqw and is non-negative on qw. Now, choose j A l 2 ðw 0 ; m W Þ with 0 e j e c and jðxþ 3 0 whenever cðxþ 3 0. Thus, Define u 0 on W 0 by y A W j f 0 on qw and j 1 0 on W 0 nqw: u 0 :¼ðL ðdþ W 0 þ aþ 1 j: As j is non-negative on qw, combining Lemma 9.1 and Lemma 9.2 shows that u 0 is nonnegative (on W 0 ). Now, define v on V by setting v equal to u on W and setting v equal to u 0 on W 0. We now investigate for each x A V the value of We consider four cases. ð~l þ aþvðxþ:

111 34 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs Case 1: x A WnqW 0. þ aþu 0 ðxþ ¼jðxÞ ¼0 by construc- Case 2: x A W 0 nqw. tion of u 0. Then, ð~l þ aþvðxþ ¼ð~L ðdþ W Then, ð~l þ aþvðxþ ¼ð~L ðdþ W 0 þ aþuðxþ e 0 by assumption on u. Case 3: x A qw 0. Lemma 9.3 with U ¼ W gives ð~l þ aþvðxþ ¼ð~L ðdþ W þ aþuðxþ P y A W 0 bðx; yþu 0 ðyþ e 0: Here, the last inequality follows as ð~l ðdþ W þ aþuðxþ e 0 by assumption on u and u0 is nonnegative. Case 4: x A qw. Lemma 9.3 with U ¼ W 0 gives ð~l þ aþvðxþ ¼ð~L ðdþ W 0 þ aþu 0 ðxþ P bðx; yþuðyþ ¼jðxÞ cðxþe0: y A W This finishes the proof. r P Proof of Corollary 1.2. Let a > 0 be given and a constant C f 0 such that bðx; yþ=mðxþ e C for all x A W. A non-negative subsolution for a þ C with respect y A VnW to the operator associated to ðb ðdþ W ; 0Þ is obviously a subsolution for a with respect to the operator associated to ðb ðdþ W ; cðdþ W Þ. By assumption such non-negative, nontrivial, bounded subsolutions for a þ C and ðb ðdþ W ; 0Þ exist for all a > 0. Therefore ðbðdþ W ; cðdþ W Þ satisfies ðsi yþ. The statement now follows from Theorem 4. r Acknowledgments. The research of M.K. is financially supported by a grant from Klaus Murmann Fellowship Programme (sdw). Part of this work was done while he was visiting Princeton University. He would like to thank the Department of Mathematics for its hospitality. He would also like to thank Jozef Dodziuk and Radek Wojciechowski for several inspiring discussions bringing up some of the questions which motivated this paper. D.L. would like to thank Andreas Weber for most stimulating discussions and Peter Stollmann for generously sharing his knowledge on Dirichlet forms on many occasions. The careful reading of the referee and partial support from German Science Foundation (DFG) are gratefully acknowledged. References [1] A. Beurling and J. Deny, Espaces de Dirichlet, I, Le cas élémentaire, Acta Math. 99 (1958), [2] A. Beurling and J. Deny, Dirichlet spaces, Proc. Natl. Acad. Sci. USA 45 (1959), [3] N. Bouleau and F. Hirsch, Dirichlet forms and analysis on Wiener space, de Gruyter Stud. Math. 14, de Gruyter, [4] F. R. K. Chung, Spectral Graph Theory, CBMS Reg. Conf. Ser. Math. 92, Amer. Math. Soc., [5] Y. Colin de Verdière, Spectres de graphes, Soc. Math. de France, Paris [6] E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge [7] E. B. Davies, Linear operators and their spectra, Cambridge Stud. Adv. Math. 106, Cambridge University Press, Cambridge 2007.

112 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs 35 [8] J. Dodziuk, Di erence equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), [9] J. Dodziuk, Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ (2006), [10] J. Dodziuk and W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, in: From local times to global geometry, control and physics, Pitman Res. Notes Math. 150 (1986), [11] J. Dodziuk and V. Matthai, Kato s inequality and asymptotic spectral properties for discrete magnetic Laplacians, in: The ubiquitous heat kernel, Contemp. Math. 398, Amer. Math. Soc. (2006), [12] W. Feller, On boundaries and lateral conditions for the Kolmogorov di erential equations, Ann. Math. (2) 65 (1957), [13] W. Feller, Notes to my paper On boundaries and lateral conditions for the Kolmogorov di erential equations, Ann. Math. (2) 68 (1958), [14] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Stud. Math. 19, de Gruyter, [15] A. Grigor yan, Analytic and geometric background of reccurrence and non-explosion of the brownian motion on riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), [16] S. Haeseler and M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, in: D. Lenz, F. Sobieczky, W. Woess, eds., Boundaries and spectra of random walks, Progr. Probab., to appear. [17] P. E. T. Jorgensen, Essential selfadjointness of the graph-laplacian, J. Math. Phys. 49 (2008). [18] M. Keller, Essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346 (2010), no. 1, [19] M. Keller and D. Lenz, Unbounded Laplacians on graphs: Basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5 (2010), no. 2. [20] Z. M. Ma and M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer, [21] B. Metzger and P. Stollmann, Heat kernel estimates on weighted graphs, Bull. Lond. Math. Soc. 32 (2000), [22] G. E. H. Reuter, Denumerable Markov processes and the associated contraction semigroups on l, Acta Math. 97 (1957), [23] P. Stollmann, A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains, Math. Z. 219 (1995), [24] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), [25] K.-T. Sturm, Analysis on local Dirichlet spaces, I, Recurrence, conservativeness and L p -Liouville properties, J. reine angew. Math. 456 (1994), [26] A. Weber, Analysis of the physical Laplacian and the heat flow on a locally finite graph, J. Math. Anal. Appl. 370 (2010), [27] J. Weidmann, Lineare Operatoren in Hilberträumen I, Teubner, Stuttgart 2000 [28] R. K. Wojciechowski, Stochastic completeness of graphs, PHD thesis, 2007, arxiv: v2. [29] R. K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J. 58 (2009), Mathematisches Institut, Friedrich Schiller Universität Jena, Jena, Germany m.keller@uni-jena.de Mathematisches Institut, Friedrich Schiller Universität Jena, Jena, Germany daniel.lenz@uni-jena.de Eingegangen 18. März 2010, in revidierter Fassung 12. Dezember 2010

113 CHAPTER 5 M. Keller, D. Lenz, Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation, Mathematical modeling of natural phenomena: Spectral Problems 5 (2010),

114 Math. Model. Nat. Phenom. Vol. 5, No. 2, 2009, pp. Unbounded Laplacians on graphs: Basic spectral properties and the heat equation Matthias Keller 1 Daniel Lenz 2 1 Mathematisches Institut, Friedrich Schiller Universität Jena, D Jena, Germany, m.keller@uni-jena.de. 2 Mathematisches Institut, Friedrich Schiller Universität Jena, D Jena, Germany, daniel.lenz@uni-jena.de, URL: Abstract. We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness. Key words: Dirichlet forms, graphs, essential self adjointness, essential spectrum, stochastic completeness AMS subject classification: 47B39, 60J27 Introduction The study of Laplacians on graphs is a well established topic of research (see e.g. the monographs [4, 6] and references therein). Such operators can be seen as discrete analogues to Schrödinger operators. Accordingly their spectral theory has received quite some attention. Such operators also arise as generators of symmetric Markov processes and they appear in the study of heat equations on discrete structures. Recently, certain themes related to unboundedness properties of such operators have become a focus of attention. These themes include definition of the operators and essential selfadjointness, absence of essential spectrum, stochastic incompleteness. Corresponding author. daniel.lenz@uni-jena.de 1

115 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs In this paper we want to survey recent developments and provide some new results. Our principle goal is to make these topics accessible to non-specialists by providing a somewhat gentle and introductory discussion. Let us be more precise. We consider a graph with weights on edges and vertices. The weights can be seen to give a generalized vertex degree. There is an obvious way to formally associate a symmetric nonnegative operator to such a graph. If the generalized vertex degrees are uniformly bounded this operator is bounded and all formal expressions make sense. If the generalized vertex degrees are not uniformly bounded already the definition of a self adjoint operator is an issue. This issue can be tackled by proving essential selfadjointness of the formal operator on the set of functions with compact support. This was done for locally finite weighted graphs by Jorgensen in [19] and for locally finite graphs by Wojciechowski in [32] and Weber in [31]. These results require local finiteness and do not allow for weights on the corresponding l 2 space. As discussed by Keller/Lenz in [21] it is possible to get rid of the local finiteness requirement and to allow for weighted spaces by using Dirichlet forms. The corresponding results give a nonnegative selfadjoint (but not necessarily essentially selfadjoint) operator in quite some generality and provide a criterion for essential selfadjointness covering the earlier results of [19, 31, 32]. These topics are discussed in Section 1. Having an unbounded nonnegative operator at ones disposal one may then wonder about its basic spectral features. These basic features include the position of the infimum of the spectrum and the existence of essential spectrum. Both issues can be approached via isoperimetric inequalities. In fact, lower bounds for the spectrum have been considered by Dodziuk [9] and Dodziuk/Kendall [11]. For planar graphs explicit estimates for the isoperimetric constant and hence for the spectrum can be found for instance in [17, 18, 22, 23, 30]. Triviality of the essential spectrum for general graphs has been considered by Fujiwara [14]. The corresponding results deal with bounded operators only. (They allow for unbounded vertex degree but then force boundedness of the operators by introducing weights on the corresponding l 2 space.) Still, the methods can be used to provide lower bounds on the spectrum and prove emptiness of the essential spectrum for unbounded Laplacians as well. For locally finite graphs this has been done by Keller in [20]. Here, we present a generalization of the results of [20] to the general setting of regular Dirichlet forms. This generalization also extends the results of [14, 11] to our setting. This is discussed in Section 5. Finally, we turn to a (possible) consequence of unboundedness in the study of the heat equation viz stochastic incompleteness. Stochastic incompleteness describes the phenomenon that mass vanishes in a diffusion process. While this may a priori not seem to be connected to unboundedness, it turns out to be connected. This has already been observed by Dodziuk/Matthai [12] and Dodziuk [10] in that they show stochastic completeness for certain bounded operators on graphs. A somewhat more structural connection is provided by our discussion below. For locally finite graphs stochastic completeness has recently been investigated by Weber in [31] and Wojciechowski [32]. In fact, Weber presents sufficient conditions and Wojciechowski gives a characterization of stochastic incompleteness. This characterization is inspired by corresponding work of Grigor yan on manifolds [16] (see work of Sturm [27] for related results as well). As shown in [21] this characterization can be extended to regular Dirichlet forms. Details are discussed in Section 8. There, we also provide some further background extending [21]. Let us mention that this circle of ideas 2

116 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs is strongly connected to questions concerning uniqueness of Markov process with given generator as discussed by Feller in [13] and Reuter in [26]. We take this opportunity to mention the very recent survey [34] of Wojciechowski giving a thorough discussion of stochastic incompleteness for manifolds and graphs (with edge weight constant to one). While our basic aim is to study unbounded Laplacians we complement our results by characterizing boundedness of the Laplacians in question in Section 3. The paper is organized as follows. In Section 1 we introduce our operators and discuss basic properties. Section 2 contains a useful minimum principle and some of its consequences. Boundedness of the Laplacians in question is characterized in Section 3. A useful tool, the so called co-area formulae are investigated in Section 4. They are used in Section 5 to provide an isoperimetric inequality which is then used to study bounds on the infimum of the (essential) spectrum. This allows us in particular to characterize emptiness of the essential spectrum. The connection to Markov processes is discussed in Section 7. A characterization of stochastic incompleteness is given in Section Graph Laplacians and Dirichlet forms Throughout V will be a countably infinite set Weighted graphs We will deal with weighted graphs with vertex set V. A symmetric weighted graph over V is a pair (b, c) consisting of a map b : V V [0, ) with b(x, x) = 0 for all x V and a map c : V [0, ) satisfying the following two properties: (b1) b(x, y) = b(y, x) for all x, y V. (b2) y V b(x, y) < for all x V. Then b is called the edge weight and c is called killing term. We consider (b, c) or rather the triple (V, b, c) as a weighted graph with vertex set V in the following way: An x V with c(x) 0 is thought to be connected to the point by an edge with weight c(x). Moreover, x, y V with b(x, y) > 0 are thought to be connected by an edge with weight b(x, y). Vertices x, y V with b(x, y) > 0 are called neighbors. More generally, x, y V are called connected if there exist x 0, x 1,..., x n V with b(x i, x i+1 ) > 0, i = 0,..., n and x 0 = x, x n = y. This allows us to define connected components of V in the obvious way. Two examples have attracted particular attention. Example (Locally finite graphs): Let (V, b, c) be a weighted graph with c 0 and b(x, y) {0, 1} for all x, y V. We can then think of the (x, y) V V with b(x, y) = 1 as connected by an edge with weight 1. The condition (b2) then implies that any x V is connected to only 3

117 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs finitely many y V. Such graphs are known as locally finite graphs. This is the class of examples studied in [20, 14, 32, 31]. Example (Locally finite weighted graphs): Let (V, b, c) be a weighted graph with c 0 and b satisfying {y : b(x, y) 0} < for all x V. Then, (V, b, c) is called a locally finite weighted graph. This is the class of examples studied in [10, 19] Dirichlet forms on countable sets Let m be a measure on V with full support (i.e., m is a map m : V (0, )). Then, (V, m) is a measure space. A particular example is given by m 1. We will deal exclusively with real valued functions. Thus, l p (V, m), 0 < p <, is defined by {u : V R : x V m(x) u(x) p < }. Obviously, l 2 (V, m) is a Hilbert space with inner product, =, m given by u, v := x V m(x)u(x)v(x) and norm u := u, u 1 2. Moreover we denote by l (V ) the space of bounded functions on V. Note that this space does not depend on the choice of m. It is equipped with the supremum norm defined by u := sup u(x). x V A symmetric nonnegative form on (V, m) is given by a dense subspace D of l 2 (V, m) called the domain of the form and a map Q : D D R with Q(u, v) = Q(v, u) and Q(u, u) 0 for all u, v D. Such a map is already determined by its values on the diagonal {(u, u) : u D} D D. This motivates to consider the restriction of Q to the diagonal as an object on its own right. Thus, for u l 2 (V, m) we then define Q(u) by { Q(u, u) : u D, Q(u) := : u D. If l 2 (V, m) [0, ], u Q(u) is lower semicontinuous Q is called closed. If Q has a closed extension it is called closable and the smallest closed extension is called the closure of Q. A map C : R R with C(0) = 0 and C(x) C(y) x y is called a normal contraction. If Q is both closed and satisfies Q(Cu) Q(u) for all normal contractions C and all u l 2 (V, m) it is called a Dirichlet form on (V, m) (see [3, 7, 15, 24] for background on Dirichlet forms). 4

118 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Let C c (V ) be the space of finitely supported functions on V. A Dirichlet form on (V, m) is called regular if its domain contains C c (V ) and the form is the closure of its restriction to the subspace C c (V ). (The standard definition of regularity for Dirichlet forms would require that D(Q) C c (V ) is dense in both C c (V ) and D(Q). As discussed in [21] this is equivalent to our definition.) 1.3. From weighted graphs to Dirichlet forms There is a one-to-one correspondence between weighed graphs and regular Dirichlet forms. This is discussed next. To the weighted graph (V, b, c) we can then associate the form Q max = Q max b,c,m : l2 (V, m) [0, ] with diagonal given by Q max (u) = 1 2 x,y V b(x, y)(u(x) u(y)) 2 + x V c(x)u(x) 2. Here, the value is allowed. Let Q comp = Q comp b,c be the restriction of Q max to C c (V ). It is not hard to see that Q max is closed. Hence Q comp is closable on l 2 (V, m) and the closure will be denoted by Q = Q b,c,m and its domain by D(Q). As discussed in [21] (see [15] as well) the following holds. Theorem 1. The regular Dirichlet forms on (V, m) are exactly given by the forms Q b,c,m with weighted graphs (b, c) over V. Remark. One may wonder whether the regularity assumption is necessary in the above theorem. It turns out that not every Dirichlet form Q max b,c,m is regular. A counterexample is provided in [21]. For a given a weighted graph (V, b, c) the different choices of measure m will produce different Dirichlet forms. Two particular choices have attracted attention. One is the choice of m 1. Obviously, this choice does not depend on b and c. Another possibility is to use n = m = m b,c given by n(x) := b(x, y) + c(x). y V The advantage of this measure is that it produces a bounded form (see below for details) Graph Laplacians Let m be a measure on V of full support, (b, c) a weighted graph over V and Q b,c,m the associated regular Dirichlet form. Then, there exists a unique selfadjoint operator L = L b,c,m on l 2 (V, m) such that D(Q) := {u l 2 (V, m) : Q(u) < } = Domain of definition of L 1/2 5

119 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs and Q(u) = L 1/2 u, L 1/2 u for u D(Q) (see e.g. Theorem in [7]). As Q is nonnegative so is L. Definition 2. Let V be a countable set and m a measure on V with full support. A graph Laplacian on V is an operator L associated to a form Q b,c,m. Our next aim is to describe the operator L more explicitly: Define the formal Laplacian L = L b,c,m on the vector space F := {u : V R : y V b(x, y)u(y) < for all x V } (1.1) by Lu(x) := 1 m(x) y V b(x, y)(u(x) u(y)) + c(x) m(x) u(x), where, for each x V, the sum exists by assumption on u. The operator L describes the action of L in the following sense. Proposition 3. Let (V, b, c) be a weighted graph and m a measure on V of full support. Then, the operator L is a restriction of L i.e., for all u D(L). D(L) {u l 2 (V, m) : Lu l 2 (V, m)} and Lu = Lu In order to obtain further information we need a stronger condition. We define condition (A) as follows: (A) For any sequence (x n ) of vertices in V such that b(x n, x n+1 ) > 0 for all n N, the equality n N m(x n) = holds. Let us emphasize that in general (A) is a condition on (V, m) and b together. However, if inf m x > 0 x V holds, then obviously (A) is satisfied for all graphs (b, c) over V. This applies in particular to the case that m 1. Given (A) we can say more about the generators [21]. Theorem 4. Let (V, b, c) be a weighted graph and m a measure on V of full support such that (A) holds. Then, the operator L is the restriction of L to D(L) = {u l 2 (V, m) : Lu l 2 (V, m)}. 6

120 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Remark. The theory of Jacobi matrices already provides examples showing that without (A) the statement becomes false [21]. The condition (A) does not imply that Lf belongs to l 2 (V, m) for all f C c (V ). However, if this is the case, then (A) does imply essential selfadjointness. In this case, Q is the maximal form associated to the graph (b, c). More precisely, the following holds [21]. Theorem 5. Let V be a set, m a measure on V with full support, (b, c) a graph over V and Q the associated regular Dirichlet form. Assume LC c (V ) l 2 (V, m). Then, D(L) contains C c (V ). If furthermore (A) holds, then the restriction of L to C c (V ) is essentially selfadjoint and the domain of L is given by D(L) = {u l 2 (V, m) : Lu l 2 (V, m)} and the associated form Q satisfies Q = Q max i.e., for all u l 2 (V, m). Q(u) = 1 2 x,y V b(x, y)(u(x) u(y)) 2 + x V c(x)u(x) 2 Remark. Essential selfadjointness may fail if (A) does not hold as can be seen by examples [21]. If inf x V m x > 0 then both (A) and LC c (V ) l 2 (V, m) hold for any graph (b, c) over V. We therefore obtain the following corollary. Corollary 6. Let V be a set and m a measure on V with inf x V m x > 0. Then, D(L) contains C c (V ), the restriction of L to C c (V ) is essentially selfadjoint and the domain of L is given by and the associated form Q satisfies Q = Q max. D(L) = {u l 2 (V, m) : Lu l 2 (V, m)} Remark. The corollary includes the case that m 1 and we recover the corresponding results of [11, 32, 31] on essential selfadjointness. (In fact, the cited works also have additional restrictions on b but this is not relevant here.) 2. Minimum principle and consequences An important tool in the proofs of the results of the previous section is a minimum principle. This minimum principle shows in particular the relevance of (A) in our considerations. This is discussed in this section. The following result is a variant and in fact slight generalization of the minimum principle from [21]. 7

121 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Theorem 7. (Minimum principle) Let (V, b, c) be a weighted graph and m a measure on V of full support. Let U V be connected. Assume that the function u on V satisfies ( L + α)u 0 on U for some α > 0, u 0 on V \ U. Then, the value of u is nonnegative in any local minimum of u. Proof. Let u attain a local minimum on U in x m. Assume u(x m ) < 0. Then, u(x m ) u(y) for all y U with b(x m, y) > 0. As u(y) 0 for y V \ U, we obtain u(x m ) u(y) 0 for all y V with b(x m, y) 0. By the super-solution assumption we find 0 b(x m, y)(u(x m ) u(y)) + c(x m )u(x m ) + m(x m )αu(x m ) 0. As b and c are nonnegative, m is positive and α > 0, we obtain the contradiction 0 = u(x m ). The relevance of (A) comes from the following consequence of the minimum principle first discussed in [21]. Proposition 8. (Uniqueness of solutions on l p ) Assume (A). Let α > 0, p [1, ) and u l p (V, m) with ( L + α)u 0 be given. Then, u 0. In particular, any u l p (V, m) with ( L + α)u = 0 satisfies u 0. Proof. We first show the first statement: Assume the contrary. Then, there exists an x 0 V with u(x 0 ) < 0. By the previous minimum principle, x 0 is not a local minimum of u. Thus, there exists an x 1 connected to x 0 with u(x 1 ) < u(x 0 ) < 0. Continuing in this way we obtain a sequence (x n ) of connected points with u(x n ) < u(x 0 ) < 0. Combining this with (A) we obtain a contradiction to u l p (V, m). As for the In particular part we note that both u and u satisfy the assumptions of the first statement. Thus, u 0. Remark. The situation for p = is substantially more complicated as can be seen by our discussion of stochastic completeness in Section 8. and in particular part (ii) of Theorem 25. Using the previous minimum principle it is not hard to prove the following result. The result is in fact true for general Dirichlet forms as can be inferred from [28, 29]. For U V we denote by Q U the closure of the Q restricted to C c (U) and by L U the associated operator. Proposition 9. (Domain monotonicity) Let (V, b, c) be a symmetric graph. Let K 1 V be finite and K 2 V with K 1 K 2 be given. Then, for any x K 1 (L K1 + α) 1 f(x) (L K2 + α) 1 f(x) for all f l 2 (V, m) with f 0 and supp f K 1. A similar statement holds for the semigroups. Proposition 10. (Convergence of resolvents/semigroups) Let (V, b, c) be a symmetric graph, m a measure on V with full support and Q the associated regular Dirichlet form. Let (K n ) be an increasing sequence of finite subsets of V with V = K n. Then, (L Kn + α) 1 f (L + α) 1 f, n for any f l 2 (K 1, m K1 ). (Here, (L Kn + α) 1 f is extended by zero to all of V.) The corresponding statement also holds for the semigroups. 8

122 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs 3. Boundedness of the Laplacian Our main topic in this paper are the consequences of unboundedness of the Laplacian. In order to understand this unboundedness it is desirable to characterize boundedness of this operator. This is discussed in this section. We start with a little trick on how to get rid of the c in certain situations. Let V be the union of V and a point at infinity. We extend a function on V to V by zero and let b(, x) = b(x, ) = c(x) for all x V. We then have b(x, y) = b(x, y) + c(x) y V y V for all x V and Q(u) = 1 2 b(x, y)(u(x) u(y)) 2 x,y V for all functions u in D(Q). We define an averaged vertex degree d = d b,c,m by ( ) d(x) := 1 b(x, y) + c(x). m(x) y V Note that d(x) = n(x)/m(x), where n was defined at the end of Section 1.3. Theorem 11. Let (V, b, c) be a weighted graph and m : V (0, ) a measure on V and L the associated formal operator. Then, the following assertions are equivalent: (i) There exists a C 0 with d(x) C for all x V. (ii) The form Q is bounded on l 2 (V, m). (iii) The restriction of L to l 2 (V, m) is bounded. (iv) The restriction of L to l (V ) is bounded. In this case the restriction of L to l p (V, m) is a bounded operator for all p [1, ] and a bound is given by 2C with C from (i). Proof. By the considerations at the beginning of the section we can assume c 0. For x V we let δ x be the function on V which is zero everywhere except in x, where it takes the value 1. The equivalence between (ii) and (iii) is obvious as the operator associated to Q is a densely defined restriction of L. Obviously (i) implies (iv) (with the bound 2C). The implication (iv)= (i) follows by considering the vectors δ x, x V. 9

123 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs (i) = (ii): As (a b) 2 2a 2 + 2b 2 we obtain Q(u, u) = 1 b(x, y)(u(x) u(y)) 2 2 x,y V b(x, y)u(x) 2 + b(x, y)u(y) 2 x,y V C x V = 2C u 2. x,y V m(x)u(x) 2 + C y V m(y)u(y) 2 Here, we used the symmetry of b and the bound (i) in the previous to the last step. (ii) = (i): This follows easily as Q(δ x, δ x ) = y V b(x, y) for all x V. It remains to show the last statement: By interpolation between l 2 and l, we obtain boundedness of the operators on l p (V, m) for p [2, ]. Using symmetry we obtain the boundedness for p [1, 2). Alternatively, we can directly establish that (i) implies the boundedness of the restriction of L on l 1 (V, m). As a bound for the operator norm on l and on l 2 is 2C, we obtain this same bound on all l p. Remark. The theorem can be seen as a generalization of the well known fact that a stochastic matrix generates an operator which is bounded on all l p. Note that the theorem gives in particular that boundedness of the operator L on l 2 (V, m) is equivalent to boundedness on l (V ). This is far from being true for all symmetric operators on l 2 (V, m). For example, let A be the operator on l 2 (N, 1) with matrix given by a x,y = 1/x if y = 1 and a x,y = 1/y if x = 1 and a x,y = 0 otherwise. Then, A is bounded on l 2 but not on l. Conversely, using e.g. the measure m(x) = x 4 on N and suitable operators with only one or two ones in each row it is not hard to construct a bounded operator on l (N) which is symmetric but not bounded on l 2 (V, m). Of course, if m is such that l 2 (V, m) is contained in l (V ) then any bounded operator on l which is symmetric (and hence closed) on l 2 must be bounded as well. 4. Co-area formulae In this section we discuss some co-area type formulae. These formulae are well known for locally finite graphs e.g. [5] and carry over easily to our setting. They are useful in many contexts as e.g. the estimation of eigenvalues via isoperimetric inequalities. We use them in this spirit as well. We start with some notation. Let (V, b, c) be a weighted graph with c 0, (which can assume without loss of generality by the trick mentioned in the beginning of Section 3.). For a subset Ω V we define Ω := {(x, y) : {x, y} Ω and {x, y} V \ Ω } 10

124 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs and Ω := 1 2 (x,y) Ω We can now come to the so called co-area formula. b(x, y). Theorem 12. (Co-area formula) Let (V, b, c) be a weighted graph with c 0. Let f : V R be given and define for t R the set Ω t := {x V : f(x) > t}. Then, 1 2 x,y V b(x, y) f(x) f(y) = Proof. For x, y V with x y we define the interval I x,y by 0 Ω t dt. I x,y := [min{f(x), f(y)}, max{f(x), f(y)}) and let I x,y be the length of the interval. Let 1 x,y be the characteristic function of I x,y. Then, (x, y) Ω t if and only if t I x,y. Thus, Ω t = 1 2 x,y V b(x, y)1 x,y (t). Thus, we can calculate 0 Ω t dt = 1 b(x, y)1 x,y (t)dt 2 0 x,y V = 1 b(x, y) 1 x,y (t)dt 2 x,y V 0 = 1 b(x, y) f(y) f(x). 2 x,y V This finishes the proof. Remark. Note that the proof is essentially a Fubini type argument. The preceding formula can be seen as a first order co-area formula as it deals with differences of functions. There is also a zeroth order co-area type formula dealing with functions themselves. This is discussed next. Theorem 13. Let V be a countable set and m : V (0, ) a measure on V. Let f : V [0, ) be given and define for t R the set Ω t := {x V : f(x) > t}. Then, m(x)f(x) = x V 0 m(ω t )dt. 11

125 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Proof. We have x Ω t if and only if 1 (t, ) (f(x)) = 1. Thus, we can calculate 0 m(ω t )dt = = 0 m(x)dt x Ω t m(x)1 (t, ) (f(x))dt 0 x V = x V m(x) = x V m(x)f(x). 0 1 (t, ) (f(x))dt This finishes the proof. 5. Isoperimetric inequalities and lower bounds on the (essential) spectrum In this section we will provide lower bound on the infimum of the (essential) spectrum using an isoperimetric inequality. This will allow us in particular to provide criteria for emptiness of the essential spectrum. Our considerations extend the corresponding parts of [9, 11, 14, 20] (as discussed in more detail below). We start with some notation used throughout this section. Let a weighted graph (V, b, c) with a measure m : V (0, ) and the associated Dirichlet form Q be given. In this setting we define the constant α(u) = α b,c,m (U) for a subset U V by α(u) = where as introduced in the previous section W = inf W U, W < x W,y W W m(w ), b(x, y) + x W c(x). Note that for a finite set W and the characteristic function 1 W of W one has Recall the definition of the normalizing measure n on V W m(w ) = Q(1 W ) 1 W 2. (5.1) n(x) = y V b(x, y) + c(x). 12

126 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Thus, we have two measures and thus two Hilbert spaces at our disposal. To avoid confusion, we will write m and n for the corresponding norms whenever necessary. Note that d(x) = n(x)/m(x). Define maximal and minimal averaged vertex degree by d U = d b,c,m (U) = inf x U d(x) and D U = D b,c,m (U) = sup d(x), x U where d is the averaged vertex degree, which was defined in Section 3. Recall d(x) = n(x)/m(x) for x V. We will also need the restrictions of operators on V to subsets of V. As in the end of Section 2denote the closure of the restriction of a closed semibounded form Q with domain containing C c (V ) to C c (U) by Q U and its associated operator by L U (for U V arbitrary). For later use we also note that for the Dirichlet form Q associated to a graph (V, b, c) with measure m on V we have inf σ(l U ) = inf u C c(u) Q(u) α(u) inf u d(x) = d 2 U x U for any U V. Here, the first equality is just the variational principle for forms, the second step follows from the definition of α and the last estimate follows by choosing W = {x} for x U. In particular, α gives upper bound on the infimum of the spectrum. It is a remarkable (and well known) fact that α > 0 implies also a lower bounds on the infimum of spectra. This is the core of the present section An isoperimetric inequality In this subsection we provide an isoperimetric inequality in our setting. This inequality (and its proof) are generalizations of the corresponding considerations of [11, 14, 20] to our setting. Proposition 14. Let (V, b, c) be a weighted graph, m : V (0, ) a measure on V and Q the associated regular Dirichlet form. Let U V and φ C c (U). Then Q(ϕ) 2 2 ϕ 2 nq(ϕ) + α b,c,m (U) 2 ϕ 4 m 0. Proof. By the trick introduced at the beginning of Section we can assume without loss of generality that c 0. Define now A by A = 1 b(x, y) ϕ(x) 2 ϕ(y) 2 = b(x, y) ϕ(x) ϕ(y) ϕ(x) + ϕ(y). 2 x,y V x,y V Following ideas of [11] for locally finite graphs (see [14, 20] as well) we now proceed as follows: By Cauchy-Schwarz inequality and a direct computation we have A 2 Q(ϕ) 1 b(x, y) ϕ(x) + ϕ(y) 2 = Q(ϕ) ( 2 ϕ 2 n 2 Q(ϕ)). x,y V 13

127 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs On the other hand we can use the first co-area formula (with f = ϕ 2 ), the definition of α and the second co-area formula to estimate A = 0 Ω t dt α Combining the two estimates on A we obtain This yields the desired result. 0 m(ω t )dt = α x V m(x)ϕ 2 (x) = α ϕ 2 m. Q(ϕ) ( 2 ϕ 2 n Q(ϕ)) ϕ 4 m Lower bounds for the infimum of the spectrum In this section we use the isoperimetric inequality of the previous section to derive bounds on the form Q. This is in the spirit of [11, 14, 20]. As usual we write a Q b (for a, b R) whenever for all u D(Q). a u 2 Q(u) b u 2 Proposition 15. Let (V, b, c) be a weighted graph, m : V (0, ) a measure on V and Q the associated regular Dirichlet form. Let U V be given and Q U the restriction of Q to U. Then, ) ( ) d U (1 1 α b,c,n (U) 2 Q U D U α b,c,n (U) 2. If D U < then furthermore D U DU 2 α b,c,m(u) 2 Q U D U + DU 2 α b,c,m(u) 2. Proof. We start by proving the first statement. Consider an arbitrary ϕ C c (U) with ϕ n = 1. Then, Proposition 14 (applied with m = n) gives Q(ϕ) 2 2Q(ϕ) + α b,c,n (U) 2 0 and hence 1 1 α b,c,n (U) 2 Q(ϕ) α b,c,n (U) 2. As this holds for all ϕ C c (U) with ϕ n = 1 and d U ϕ m ϕ n D U ϕ m by definition of d U and D U, we obtain the first statement. 14

128 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs We now turn to the last statement. By definition of D U we have ϕ n D U ϕ m. Thus, Proposition 14 gives Q(ϕ) 2 2D U ϕ 2 mq(ϕ) + α b,c,m (U) 2 ϕ 4 m 0. Considering now ϕ C c (U) with ϕ m = 1 we find that D U DU 2 α b,c,m(u) Q(ϕ) D U + DU 2 α b,c,m(u) for all such ϕ. This finishes the proof. As a first consequence of the previous proposition we obtain the following corollary first proven for m = n, and locally finite graphs in [14]. Corollary 16. For a weighted graph (V, b, c) and m = n we obtain 1 1 α 2b,c,n Q αb,c,n 2. A second consequence of the above proposition is that the bottom of the spectrum being zero can be characterized by the constant α in the case of bounded operators. This is our version of the well known result that a graph with finite vertex degree is amenable if and only if zero belongs to the spectrum of the corresponding Laplacian. Corollary 17. Let (V, b, c) be a weighted graph and D U < for U V. Then inf σ(l U ) = 0 if and only if α b,c,m (U) = 0. Proof. The direction = follows from Proposition 15 and the other direction = follows directly from equation 5.1. Remark. The direction = in the previous corollary does not depend on the assumption D U < for U V and is true in general Absence of essential spectrum In this subsection we use the results of the previous subsection to study absence of essential spectrum. The key idea is that the essential spectrum of an operator is a suitable limit of the spectra of restrictions going to infinity. This reduces the problem of proving absence of essential spectrum to proving lower bounds on the spectrum at infinity. For locally finite graphs this has been done in [14, 20]. Let (V, b, c) be a weighted graph. Let K be the set of finite sets in V. This set is directed with respect to inclusion and hence a net. Limits along this net will be denoted by lim K K and we will say that K tends to V. We then define α b,c,m ( V ) = lim K K α b,c,m (V \ K). 15

129 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Likewise let d V = d b,c,m ( V ) = lim d b,c,m (V \ K), K K D V = D b,c,m ( V ) = lim D b,c,m (V \ K). K K The following proposition is certainly well known and has in fact already been used in the past (see e.g. [20]). We include a proof as we could not find one in the literature. Note also that our result is more general than the result mentioned e.g. in [20] as we deal with forms. Proposition 18. Let Q be a closed form on l 2 (V, m), whose domain of definition contains C c (V ). If Q is bounded below then and if Q is bounded above then inf σ ess (B) = lim K K inf σ(b V \K ). sup σ ess (B) = lim K K sup σ(b V \K ) holds, where B is the operator associated to Q and B V \K the operator associated to Q V \K for finite K V. Proof. It suffices to show the statement for Q which are bounded below (as the other statement then follows after replacing Q by Q). Without loss of generality we can assume Q 0. Let λ 0 := inf σ ess (B). As the essential spectrum does not change by finite rank perturbations we have σ ess (B) = σ ess (B V \K ) σ(b V \K ) and hence λ 0 σ(b V \K ) for any finite K V. This gives inf σ ess (B) lim K K inf σ(b V \K ). To show the opposite inequality it suffices to prove that for arbitrary λ < λ 0 we have inf σ(b V \K ) > λ for all sufficiently large finite K. Fix λ 1 with λ < λ 1 < λ 0 and choose δ > 0 such that λ + δ < λ 1. Moreover let ε = λ 1 (λ + δ). λ The spectral projection E (,λ1 ] of B to the interval (, λ 1 ] is a finite rank operator since B 0. This easily implies lim K K E (,λ 1 ]P K = 0, 16

130 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs where P K is the projection onto l 2 (V \ K, m). Thus, there is K ε finite with for all K K ε finite. In particular, we have E (,λ1 ]P K 2 ε E (,λ1 ]ψ 2 ε (5.2) for all ψ l 2 (V \ K ε, m) with ψ = 1 (as for such ψ we have ψ = P Kε ψ). Consider now a finite K with K K ε and let ψ l 2 (V \ K, m) be given with ψ = 1 such that Q(ψ) = Q V \K (ψ) (inf σ(b V \K ) + ε). Let ρ ψ ( ) be the spectral measure associated to B and ψ. Then Q(ψ) = 0 λ 1 λ 1 tdρ ψ (t) t dρ ψ (t) λ 1 dρ ψ (t) = λ 1 ( ψ, ψ E (,λ1 ]ψ, E (,λ1 ]ψ ) λ 1 (1 ε). In the first step we used that B is positive and in the last step we used (5.2). By our choice of ψ and ε we get inf σ(b V \K ) Q(ψ) ε λ 1 (1 ε) ε = λ + δ > λ. This finishes the proof. Combining this proposition with Proposition 15 one gets estimates for the essential spectrum of the operator L. The following provides a generalization of a main result of Fujiwara s theorem [14] to our setting. Fujiwara s result deals with locally finite graphs and m = n. Theorem 19. Let (V, b, c) be a weighted graph, m : V (0, ) a measure on V and Q the associated regular Dirichlet form. Assume D V = D b,c,m ( V ) <. Then, σ ess (L) = {D V } if and only if α b,c,m ( V ) = D V. Proof. One direction = follows directly from Proposition 15 and Proposition 18. The other direction = follows from inf σ(l U ) α b,c,m (U) D b,c,m (U) for U V and Proposition 18 by taking U = V \ K for K finite and considering the limit for K tending to V. 17

131 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Remark. The assumption D b,c,m ( V ) < implies boundedness of the operator (see Section 3.). Thus, σ ess (L) must be non-empty in this case. Proposition 18 shows that inf σ(l V \K ) and sup σ(l V \K ) converge necessarily to points in the essential spectrum of L (for K tending to V ). The only way how the essential spectrum can consist of only one point is then that both limits agree. As inf σ(l V \K ) α(v \ K) and sup σ(l V \K ) D b,c,m this is only possible for α b,c,m ( V ) = D V. In this way the theorem characterizes the only way how essential spectrum can consist of only one point. The next theorem is a generalization to our setting of Theorem 2 in [20], which deals with locally finite graphs and m 1. Theorem 20. Let (V, b, c) be a weighted graph, m : V (0, ) a measure on V and Q the associated regular Dirichlet form. Assume α b,c,n > 0. Then σ ess (L) = if and only if d V =. Proof. One direction = follows directly from Proposition 15 and 18. The other direction = follows from the fact that for all U V we have inf σ(l U ) d b,c,m (U) and Proposition An application In this section we consider a locally finite graph i.e., (V, b, 0) with b taking values in {0, 1} with the measure m 1. Let Q 0 be the associated form and the associated operator. Let c : V [0, ) be given and define L to be the operator associated to Q b,c,m. Thus, L = + c (at least on the formal level). This decomposition of L leads to a similar decomposition of the parameters α. In this way, both the geometry (encoded by b) and the potential (encoded by c) can lead to absence of essential spectrum according to the preceding considerations. This is discussed in further details next. The Cheeger constant β U of a subset U V is the smallest number such that for all finite W U W β U vol(w ), where W = 1 W, 1 W = x W,y / W b(x, y) is defined as above and vol(w ) = 1 W 2 n = x W n(x). If β V > 0 one says that the graph is hyperbolic. Furthermore, let γ U be given as the smallest number such that for all finite W U c(w ) γ U vol(w ), where c(w ) = c1 W, 1 W = x W c(x). For example γ V > 0, if there is C > 0 such that c(x) Cd(x), where d(x) is the vertex degree. Finally let β V = lim β V \K and γ V = lim γ V \K. K K V K Hence the preceding section immediately gives the following corollary of Theorem

132 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs Corollary 21. Let β V > 0 or γ V > 0. Then σ ess (H) = if and only if d(x n ) + c(x n ) along any infinite path (x n ) with pairwise distinct vertices. 7. Graph Laplacians and Markov processes We have already discussed that our Laplacians come from Dirichlet forms. Now, Dirichlet forms and symmetric Markov processes are intimately connected. The crucial link is given by the semigroup generated by a Dirichlet form. The connection to Markov processes means that there is a wealth of results on the semigroup associated to a graph Laplacian, there is a good interpretation of properties of the semigroup in terms of a stochastic process. Details are discussed in this section Graph Laplacians, their semigroup and the heat equation Let a measure m on V with full support and a graph (b, c) over V be given. Let Q be the associated form and L its generator. Standard theory [8, 15, 24] implies that the operators of the associated semigroup e tl, t 0, and the associated resolvent α(l+α) 1, α > 0 are positivity preserving and even markovian. Positivity preserving means that they map nonnegative functions to nonnegative functions. Markovian means that they map nonnegative functions bounded by one to nonnegative functions bounded by one. This can be used to show that semigroup and resolvent extend to all l p (V, m), 1 p. These extensions are consistent i.e., two of them agree on their common domain [7]. The corresponding generators are denoted by L p, in particular L = L 2. We can describe the action of the operator L p explicitly. More precisely, the situation on l 2 (see Proposition 3 and Theorem 4) holds here as well: Theorem 22. Let (V, b, c) be a weighted graph and m a measure on V of full support. Then, the operator L p is a restriction of L for any p [1, ]. If furthermore (A) holds, then the operator L is the restriction of L to {u l p (V, m) : Lu l p (V, m)}. A function N : [0, ) V R is called a solution of the heat equation if for each x V the function t N t (x) is continuous on [0, ) and differentiable on (0, ) and for each t > 0 the function N t belongs to the domain of L, i.e., the vector space F and the equality d dt N t(x) = LN t (x) holds for all t > 0 and x V. For a bounded solution N validity of this equation can easily be seen to automatically extend to t = 0 i.e., t N t (x) is differentiable on [0, ) and d dt N t(x) = LN t (x) holds for any t 0. 19

133 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs The following theorem is a standard result in the theory of semigroups. A proof in our context can be found in [21] (see [32, 33, 31] for related material on special graphs). Theorem 23. Let L be a selfadjoint restriction of L, which is the generator of a Dirichlet form on l 2 (V, m). Let v be a bounded function on V and define N : [0, ) V R by N t (x) := e tl v(x). Then, the function N(x) : [0, ) R, t N t (x), is differentiable and satisfies for all x V and t 0. d dt N t(x) = LN t (x) Let us conclude this section by noting that the semigroups are positivity improving for connected graphs. This has been shown in [21] in our setting after earlier results in [8, 31, 32] for locally finite graphs. Theorem 24. (Positivity improving) Let (V, b, c) be a connected graph and L be the associated operator. Then, both the semigroup e tl, t > 0, and the resolvent (L + α) 1, α > 0, are positivity improving (i.e., they map nonnegative nontrivial l 2 -functions to strictly positive functions) Connection to Markov processes In this section we discuss the relationship between Dirichlet forms and Markov processes in our context. Let Q be the Dirichlet form associated to a weighted graph (V, b, c) with measure m. For convenience we assume m 1. Let L be the associated operator and e tl, t > 0, the associated semigroup. We will take the point of view that we already know that e tl is a semigroup of transition properties of a Markov process. We will then show how we can identify the key quantities of the Markov process in terms of the graph (V, b, c). A (time homogenous) Markov process on V consists of a particle moving in time without memory between the points of V. It is characerized by two sets of quantities: These are a function a : V [0, ) such that e tax is the probability that a particle in x at time 0 is still in x at time t. a function q : V V [0, ) such that q x (y) is the probability that the particle jumps to y from x. Given such a Markov process we can define P t (x, y) := Probability that the particle is in y at time t if it starts in x at time 0 for t 0, x, y V and the operators P t provide a semigroup of operators. It is then possible to infer the quantities a and q from the behavior of P t for small t in the following way: P t (x, x) is the probability to find the particle at x at time t (for a particle starting at x at time 0). This means that the particle has either stayed at x for the whole time between 0 and t or has jumped 20

134 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs from x away and come back by the time t. The probability that the particle stayed in x (i.e., did not move away) is e tax. The event that the particle left x and returned by the time t means that the particle left x, which occurs with probability 1 e tax, and then returned from V \ {x} to x in the remaining time, which occurs with probability r(t) going to zero for t 0. Accordingly we have P t (x, x) = e tax + φ x (t), where φ x summarizes the probability of returning to x, is therefore bounded by (1 e tax )r(t) and hence has derivative equal to zero at t = 0. We therefore obtain d dt P t (x, x) = a x + φ x(0) = a x. t=0 By a similar reasoning the probability P t (x, y) is governed by the event that the particle starts at x at time 0 and has done one jump to y and then stayed in y up to the time t. The probability p t for this event satisfies (1 e tax )q x (y)e tay p t (1 e tax )q x (y). Here, the term e tay serves to take into account that the particle did not leave y. Accordingly, P t (x, y) = p t + ψ(t), where the derivative of ψ at 0 is zero and we obtain d dt P t (x, y) = a x q x (y) + ψ (0) = a x q x (y). t=0 We now return to the Dirichlet form setting. As e tl describes a Markov process we can now set P t (x, y) = e tl δ x, δ y for t 0, x, y V and use this to calculate the the a s and q s in terms of b and c as follows: b(x, y) + c(x) = Q(δ x, δ x ) = d dt e tl δ x, δ x = d t=0 dt P t (x, x) = a x t=0 y V and b(x, y) = Q(δ x, δ y ) = d dt e tl δ x, δ y = d t=0 dt P t (x, y) = q x (y)a x. t=0 This gives b(x, y) q x (y) = z V b(x, z) + c(x), a x = b(x, z) + c(x) z V for all x, y V. Note that symmetry of b does not imply symmetry of q but rather a x q x (y) = a y q y (x). If m is not identically equal to one, we will have to normalize the formula for P above by setting 1 P t (x, y) = m(x)m(y) e tl δ x, δ y and change the emerging formulae accordingly. 21

135 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs 8. Stochastic completeness We consider a Dirichlet form Q on a weighted graph (V, b, c) with associated operator L and semigroup e tl. The preceding considerations show that 0 e tl 1(x) 1 for all t 0 and x V. The question, whether the second inequality is actually an equality has received quite some attention. In the case of vanishing killing term, this is discussed under the name of stochastic completeness or conservativeness. In fact, for c 0 and b(x, y) {0, 1} for all x, y V, there is a characterization of stochastic completeness of Wojciechowski [32] (see the introduction for discussion of related results of Feller [13] and Reuter [26] as well). This characterization is an analogue to corresponding results on manifolds of Grigor yan [16] and results of Sturm for general strongly local Dirichlet forms [27]. Our first main result concerns a version of this result for arbitrary regular Dirichlet forms on graphs. As we allow for a killing term c we have to replace e tl 1 by the function M t (x) := e tl 1(x) + t 0 e sl c(x)ds, x V. It is possible (and necessary) to show that this quantity is well defined. In fact, it can be proven that it satisfies 0 M 1 and that for each x V, the function t M t (x) is continuous and even differentiable [21]. Note that for c 0, M = e tl 1 whereas for c 0 the inequality M t > e tl 1 holds on any connected component of V on which c does not vanish identically (as the semigroup is positivity improving). We can give an interpretation of M in terms of a diffusion process on V as follows: For x V, let δ x be the characteristic function of {x}. A diffusion on V starting in x with normalized measure is then given by 1 m(x) δ x at time t = 0. It will yield to the amount of heat δ x e tl δ x m(x), 1 = m(x), e tl 1 = e tl (x, y) = e tl 1(x) y V within V at the time t. Thus, the first term of M describes the amount of heat within the graph at a given time. Moreover, at each time s the rate of heat killed at the vertex y by the killing term c is given by c(y)e sl (x, y). The total amount of heat killed at y till the time t is then given by t 0 c(y)e sl (x, y)ds. The total amount of heat killed at all vertices by c till the time t is accordingly given by y V t 0 c(y)e sl (x, y)ds = t 0 e sl (x, y)c(y)ds = y V t 0 (e sl c)(x)ds. Thus, the second term of M describes the total amount of heat killed up to time t within the graph. Altogether, 1 M t is then the amount of heat transported to the boundary of the graph by the time 22

136 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs t and M t can be interpreted as the amount of heat, which has not been transported to the boundary of the graph at time t. Our question concerning stochastic completeness then becomes whether the quantity 1 M t vanishes identically or not. Our result reads (see [21] for a proof): Theorem 25. (Characterization of heat transfer to the boundary) Let (V, b, c) be a weighted graph and m a measure on V of full support. Then, for any α > 0, the function w := 0 αe tα (1 M t )dt satisfies 0 w 1, solves ( L + α)w = 0, and is the largest nonnegative function l 1 with ( L + α)l 0. In particular, the following assertions are equivalent: (i) For any α > 0 there exists a nontrivial, nonnegative, bounded l with ( L + α)l 0. (ii) For any α > 0 there exists a nontrivial, bounded l with ( L + α)l = 0. (iii) For any α > 0 there exists an nontrivial, nonnegative, bounded l with ( L + α)l = 0. (iv) The function w is nontrivial. (v) M t (x) < 1 for some x V and some t > 0. (vi) There exists a nontrivial, bounded, nonnegative N : V [0, ) [0, ) satisfying LN + d dt N = 0 and N 0 0. Let us give a short interpretation of the conditions appearing in the theorem. Conditions (i), (ii) and (iii) deal with eigenvalues of L considered as an operator on l (V ). Thus, they concern spectral theory in l (V ). Condition (v) refers to loss of mass at infinity. Finally condition (vi) is about unique solutions of a partial difference equation. Thus, the result connects properties from stochastic processes, spectral theory and partial difference equations. Sketch of proof. We refrain from giving a a complete proof of the theorem but rather discuss three key elements of the proof and how they fit together. These are the following three steps: (S1) If N : [0, ) V R is a bounded solution of d dt N = LN, then v = 0 αe tα N t dt is a solution to ( L + α)v = 0 for any α > 0. (S2) The function N = 1 M satisfies 0 N 1 and d dt N = LN. (S3) The function w = 0 αe tα (1 M t )dt is the largest solution of ( L + α)v = 0 with 0 v 1. 23

137 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs The proof of the first step is a direct calculation via partial integration. The second step is a direct calculation but requires quite some care as the quantities are defined via sums and integrals whose convergence is not clear. The fact that w of the last step is a solution follows from the second step. The minimality of the solution requires some care. It follows by approximating the graph via finite graphs. Here, a nontrivial issue is that this approximation may actually cut infinitely many edges (as we do not have locally finite edge degree). Given the three steps, the proof of the theorem goes along the following line: The implication (v) = (i) follows from Step (S1) and (S2). The implication (i) = (v) follows from the maximality property in Step (3). The implication (v) = (vi) follows from Step (S2). The implication (vi) = (v) follows from Step (S1). The equivalence between (iv) and (v) is immediate from Step (S3). The equivalence between (i), (ii) and (iii) follows by taking suitable minima of (super-) solutions. Definition 26. The weighted graph (V, b, c) is said to be stochastically complete if one of the equivalent assertions of the theorem holds. Corollary 27. Assume the situation of the previous theorem. Let L be the operator associated to the graph (V, b, c). If L gives rise to a bounded operator on l 2 (V ), then the graph (V, b, c) is stochastically complete. Proof. If L is bounded on l 2 (V, m) it is bounded on l (V ) by Theorem 11. Then, the spectrum of L on l is bounded and hence its set of eigenvalues is bounded as well. Thus, (ii) of the theorem must fail (for large α). Remark. (a) The corollary shows that stochastic completeness is a phenomenon for unbounded operators. (b) The corollary generalizes the results of Dodziuk/Matthai [12] and Wojciechowski [32]. It is furthermore relevant as its proof gives an abstract i.e., spectral theoretic reason for stochastic completeness in the case of bounded operators. Let us finish this section by discussing how the existence of α > 0 and t > 0 and x V with certain properties in the above theorem is actually equivalent to the fact that all α > 0, t > 0 and x V have these properties. We first discuss the situation concerning the α s. Proposition 28. Let (V, b, c) be a weighted graph and m a measure on V of full support. Then, the following are equivalent: (i) For any α > 0 there exists a nontrivial, nonnegative, bounded l with ( L + α)l 0. (ii) For some α > 0 there exists a nontrivial, nonnegative, bounded l with ( L + α)l 0. Proof. It suffices to show the implication (ii) = (i): By the maximality property of the function w = αe tα (1 M 0 t )dt discussed in the third step of the proof of the main result, (ii) implies that M t (x) < 1 for some x V and t > 0. Now, the claim (i) follows from the second step discussed in the proof of the main result. 24

138 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs We now show that loss of mass in one point at one time is equivalent to loss of mass in all points at all times (if the graph is connected). For locally finite graphs this is discussed in [32]. Proposition 29. Let (V, b, c) be a connected weighted graph and m a measure on V of full support. Let M be defined as above. Then, the following assertions are equivalent: (i) There exist x V and t > 0 with M t (x) < 1. (ii) M t (x) < 1 for all x V and all t > 0 Proof. The implication (ii) = (i) is clear. It remains to show the reverse implication. A direct calculation (invoking t+s...dr = s...dr + t+s...dr) shows that 0 0 s This easily gives that M t+s = e sl M t + s 0 e rl c dr. (1) M t 1 for some t > 0 implies M nt 1 for all n N. (2) M t 1 for some t > 0 implies that M t+s < 1 for all s > 0. (Here (1) follows by induction and (2) follows as M t 1 implies M t 1 and M t (x) < 1 for some x V. As the graph is connected this implies e sl M t < e sl 1 and the statement follows.) Assume now that M t (x) < 1 for some x V and t > 0. We consider M r for r > t and for r < t separately: By (2), M r < 1 for all r > t. Assume that M r = 1 for some 0 < r < t, then M s = 1 for all s r by (2). Hence, by (1) M ns = 1 for all n N and 0 < s r. This gives M r = 1 for all r > 0 which contradicts M t 1. Thus, M r 1 for all 0 < r < t. Hence, by (2) M r < 1 for all 0 < r t. Acknowledgements. It is our great pleasure to acknowledge fruitful and stimulating discussions with Peter Stollmann, Radek Wojciechowski, Andreas Weber and Jozef Dodziuk on the topics discussed in the paper. References [1] A. Beurling, J. Deny. Espaces de Dirichlet. I. Le cas élémentaire. Acta Math., 99 (1958), [2] A. Beurling, J. Deny. Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A., 45 (1959), [3] N. Bouleau, F. Hirsch. Dirichlet forms and analysis on Wiener space. Volume 14 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, [4] F. R. K. Chung. Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, 92, American Mathematical Society, Providence, RI,

139 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs [5] F. R. K. Chung, A. Grigoryan, S.-T. Yau. Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs. Comm. Anal. Geom., 8 (2000), No. 5, [6] Y. Colin de Verdière. Spectres de graphes. Soc. Math. France, Paris, [7] E. B. Davies. Heat kernels and spectral theory. Cambridge University press, Cambridge, [8] E. B. Davies. Linear operators and their spectra. Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, [9] J. Dodziuk. Difference Equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc., 284 (1984), No. 2, [10] J. Dodziuk. Elliptic operators on infinite graphs. Analysis, geometry and topology of elliptic operators, , World Sci. Publ., Hackensack, NJ, [11] J. Dodziuk, W. S. Kendall. Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), 68 74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, [12] J. Dodziuk, V. Matthai. Kato s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel, 69 81, Contemp. Math., 398, Amer. Math. Soc., Providence, RI, [13] W. Feller. On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. of Math. (2), 65 (1957), [14] K. Fujiwara. Laplacians on rapidly branching trees. Duke Math Jour., 83 (1996), No. 1, [15] M. Fukushima, Y. Oshima, M. Takeda. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, [16] A. Grigor yan. Analytic and geometric background of reccurrence and non-explosion of the brownian motion on riemannian manifolds. Bull. Amer. Math. Soc. (N.S.), 36 (1999), No. 2, [17] O. Häggström, J. Jonasson, R. Lyons. Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab., 30 (2002), No. 1, [18] Y. Higuchi, T. Shirai. Isoperimetric constants of (d, f)-regular planar graphs. Interdiscip. Inform. Sci., 9 (2003), No. 2, [19] P. E. T. Jorgensen. Essential selfadjointness of the graph-laplacian. J. Math. Phys., 49 (2008), No. 7, , 33p. 26

140 Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs [20] M. Keller. The essential spectrum of Laplacians on rapidly branching tesselations. Math. Ann., 346 (2010), No. 1, [21] M. Keller, D. Lenz. Dirichlet forms and stochastic completeness of graphs and subgraphs. preprint 2009, arxiv: [22] M. Keller, N. Peyerimhoff. Cheeger constants, growth and spectrum of locally tessellating planar graphs. to appear in Math. Z., arxiv: [23] B. Mohar. Light structures in infinite planar graphs without the strong isoperimetric property. Trans. Amer. Math. Soc., 354 (2002), No. 8, [24] Z.-M. Ma and M. Röckner. Introduction to the theory of (non-symmetric) Dirichlet forms. Springer-Verlag, Berlin, [25] B. Metzger, P. Stollmann. Heat kernel estimates on weighted graphs. Bull. London Math. Soc., 32 (2000), No. 4, [26] G. E. H. Reuter. Denumerable Markov processes and the associated contraction semigroups on l. Acta Math., 97 (1957), [27] K.-T. Sturm. textitanalysis on local Dirichlet spaces. I: Recurrence, conservativeness and L p -Liouville properties. J. Reine Angew. Math., 456 (1994), No [28] P. Stollmann. A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains. Math. Z., 219 (1995), No. 2, [29] P. Stollmann, J. Voigt. Perturbation of Dirichlet forms by measures. Potential Anal. 5 (1996), No. 2, [30] H. Urakawa. The spectrum of an infinite graph. Canad. J. Math., 52 (2000), No. 5, [31] A. Weber. Analysis of the physical Laplacian and the heat flow on a locally finite graph. Preprint (2008), arxiv: [32] R. K. Wojciechowski. Stochastic completeness of graphs, PhD thesis, arxiv: v2. [33] R. K. Wojciechowski. Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J., 58 (2009), No. 3, [34] R. K. Wojciechowski. Stochastically Incomplete Manifolds and Graphs. Preprint 2009, arxiv:

141 CHAPTER 6 M. Keller, D. Lenz, R. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Mathematische Zeitschrift 274 (2013),

142 Math. Z. (2013) 274: DOI /s Mathematische Zeitschrift Volume growth, spectrum and stochastic completeness of infinite graphs Matthias Keller Daniel Lenz Radosław K. Wojciechowski Received: 16 February 2012 / Accepted: 16 September 2012 / Published online: 16 November 2012 Springer-Verlag Berlin Heidelberg 2012 Abstract We study the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs. For a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum. We then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs. Mathematics Subject Classification (2000) Primary 39A12; Secondary 58J35 1 Introduction The aim of this paper is to investigate the connections between volume growth, uniqueness of bounded solutions for the heat equation, and spectral properties for infinite weighted graphs. To do so, we proceed in two steps. We first establish these connections on a class of graphs with a very weak spherical symmetry. We then give some comparison results for general graphs by using a notion of curvature. A very general framework for studying operators on discrete measure spaces was recently established in [37]. We use this set up throughout with the additional assumption that the underlying weighted graphs are locally finite. M. Keller D. Lenz Mathematisches Institut, Friedrich Schiller Universität Jena, Jena, Germany m.keller@uni-jena.de D. Lenz daniel.lenz@uni-jena.de R. K. Wojciechowski (B) York College of the City University of New York, Jamaica, NY 11451, USA rwojciechowski@gc.cuny.edu 123

143 906 M. Keller et al. We first introduce the class of weakly spherically symmetric graphs which, compared with spherically symmetric graphs, can have very little symmetry. Still, these graphs turn out to be accessible to a detailed analysis much as those with a full spherical symmetry. More precisely, we can characterize them by the fact that: Their heat kernels are spherically symmetric (Theorem 1 in Sect. 3). Moreover, for operators arising on these graphs we prove: An explicit estimate for the bottom of their spectrum and a criterion for the discreteness of the spectrum in terms of volume growth and the boundary of balls (Theorem 3 in Sect. 4). A characterization of stochastic completeness in terms of volume growth and the boundary of balls (Theorem 5 in Sect. 5). Both the estimate for the bottom of the spectrum and the condition for stochastic completeness involve the ratio of a generalized volume of a ball to its weighted boundary. In this sense, our estimates complement the classic lower bound on the bottom of the spectrum of the standard graph Laplacian in terms of Cheeger s constant given by Dodziuk in [15], in the case of unbounded geometry. These results give rise to examples of graphs of polynomial volume growth which have positive bottom of the spectrum and are stochastically incomplete. Therefore, as a surprising consequence, in the standard graph metric there are no direct analogues to the theorems of Grigor yan, relating volume growth and stochastic completeness [25], and of Brooks, relating volume growth and the bottom of the essential spectrum [9], from the manifold setting. For stochastic completeness this was already observed in [57]. These examples are studied at the end of the paper, in Sect. 6. We now turn to the second step of our investigation, i.e., the comparison of general weighted graphs to weakly spherically symmetric ones. In this context, we provide: heat kernel comparisons (Theorem 2 in Sect. 3) comparisons for the bottom of the spectrum (Theorem 4 in Sect. 4) comparisons for stochastic completeness (Theorem 6 in Sect. 5). The heat kernel comparison, Theorem 2, is given in the spirit of results of Cheeger and Yau [12]. However, in contrast to other works done for graphs in this area, e.g. [51,56], as we use weakly spherically symmetric graphs, we require very little symmetry for our comparison spaces. The comparisons are then done with respect to a certain curvature (and, in the case of stochastic completeness, also to potential) growth. We combine these inequalities with an analogue to a theorem of Li [41], which was recently proven in our setting in [28,39], to obtain comparisons for the bottom of the spectrum, Theorem 4. The spectral comparisons give some analogues to results of Cheng [13] and extend inequalities found for graphs in [10,52,58]. The comparison results for stochastic completeness, Theorem 6, are inspired by work of Ichihara [34] and are found in Sect. 5. The article is organized as follows: in the next section we introduce the set up. This is followed by the heat kernel theorems, Theorems 1 and 2, in Sect. 3, the spectral estimates, Theorems 3 and 4, in Sect. 4 and the considerations about stochastic completeness, Theorems 5 and 6, in Sect. 5. The proofs are given within each section. In Sect. 6, wediscuss the applications to standard graph Laplacians and give the examples of polynomial volume growth announced above. Finally, in Appendix A, we prove some general facts concerning commuting operators which are used in the proof of Theorem

144 Volume growth, spectrum and stochastic completeness The set up and basic facts Our basic set up, which is included in [37,38], is as follows: let V be a countably infinite set and m : V (0, ). Extending m to all subsets of V by countable additivity gives a measure of full support and (V, m) is then a measure space. The map b : V V [0, ), which characterizes the edges, is symmetric and has zero diagonal. If b(x, y) >0, then we say that x and y are neighbors, writing x y, and think of b(x, y) as the weight of the edge connecting x and y. Moreover, let c : V [0, ) be a map which we call the potential or killing term. Ifc(x) >0, then we think of x as being connected to an imaginary vertex at infinity with weight c(x). We call the quadruple (V, b, c, m) a weighted graph. Whenever c 0, we denote the weighted graph (V, b, 0, m) as the triple (V, b, m). If, furthermore, b : V V {0, 1}, then we speak of (V, b, m) as a standard graph. We say that a weighted graph is connected if, for any two vertices x and y, there exists a sequence of vertices (x i ) i=0 n such that x 0 = x, x n = y and x i x i+1 for i = 0, 1,...,n 1. We say that a weighted graph is locally finite if every x V has only finitely many neighbors, i.e., b(x, y) vanishes for all but finitely many y V. Throughout the paper we assume that all weighted graphs (V, b, c, m) in question are connected and locally finite. In this setting, weighted graph Laplacians and Dirichlet forms on discrete measure spaces were recently studied in [37], whose notation we closely follow (see also the seminal work [6] on finite graphs and [24] for background on general Dirichlet forms). Let C(V ) denote the set of all functions from V to R and let { l 2 (V, m) = f C(V ) } f 2 (x)m(x) < x V denote the Hilbert space of functions square summable with respect to m with inner product given by f, g = x V f (x)g(x)m(x). We then define the form Q with domain of definition D(Q) = C c (V ), Q, where C c (V ) denotes the space of finitely supported functions, the closure is taken with respect to, Q :=, + Q(, ) in l 2 (V, m), andq acts by Q( f, g) = 1 2 x,y V b(x, y) ( f (x) f (y))(g(x) g(y)) + x V c(x) f (x)g(x). Such forms are regular Dirichlet forms on the measure space (V, m), see[37]. By general theory (see, for instance [24]) there is a selfadjoint positive operator L with domain D(L) l 2 (V, m) such that Q( f, g) = Lf, g for f D(L) and g D(Q). By [37, Theorem 9] we know that L is a restriction of the formal Laplacian L which acts as L f(x) = 1 b(x, y) ( f (x) f (y)) + c(x) m(x) m(x) f (x) y V 123

145 908 M. Keller et al. and, as (V, b, c, m) is locally finite, is defined for all functions in C(V ). Alternatively, as shown in[37] as well, one can consider L to be an extension the so-called Friedrichs extension of the operator L 0 defined on C c (V ) by L 0 g = Lg. Two very prominent examples from the standard setting are the graph Laplacian given by the additional assumptions that b : V V {0, 1}, c 0, and m 1sothat f (x) = ( f (x) f (y)) y x and the normalized graph Laplacian given by the additional assumptions that b : V V {0, 1}, c 0, and m deg so that f (x) = 1 ( f (x) f (y)). deg(x) y x Here, deg(x) = {y y x} for x V,where { } denotes the cardinality of a set, is finite for all x V by the local finiteness assumption. In the following sections, we will compare our results to those for and found in the literature. Furthermore, we will illustrate some of our results for the operator at the end of the paper in Sect. 6. We note that is bounded on l 2 (V ) := l 2 (V, 1) if and only if deg is bounded, while is always bounded on l 2 (V, deg). We will often fix a vertex x 0 and consider spheres and balls S r = S r (x 0 ) ={x d(x, x 0 ) = r} and B r = B r (x 0 ) = r S i (x 0 ) around x 0 of radius r. Here, d(x, y) is the usual combinatorial metric on graphs, that is, the number of edges in the shortest path connecting x and y. Definition 2.1 The outer and inner curvatures κ ± : V [0, ) are given by κ ± (x) = 1 b(x, y) for x S r. m(x) y S r±1 Remark We refer to these quantities as curvatures as, for c 0, Ld(x 0, ) = κ ( ) κ + ( ) is often referred to as a curvature-type quantity for graphs, see [17,33,54]. Moreover, in light of our Theorem 1 and [12, Proposition 2.2], as well as our comparison results, Theorems 2, 4, and6, and their counterparts in the manifold setting which can be found in [12,13,34], it seems reasonable to relate κ ± to a type of curvature on a manifold. Several other notions of curvature have been introduced for planar graphs [4,5,31], cell complexes [21], and general metric measure spaces [8,43,45,49,50]. See also the recent work on Ricci curvature of graphs [3,35,42]. It is not presently clear how these different notions of curvature are related. Definition 2.2 We call a function f : V R spherically symmetric if its values depend only on the distance to x 0, i.e., if f (x) = g(r) for x S r (x 0 ) for some function g defined on N 0 ={0, 1, 2,...}. In this case, we will often write f (r) for f (x) whenever x S r (x 0 ) and set, for convenience, f ( 1) = i=0

146 Volume growth, spectrum and stochastic completeness 909 Let à be the operator on C(V ) that averages a function over a sphere around x 0, i.e., (Ãf)(x) = 1 f (y)m(y) m(s r ) for f : V R and x S r. This operator is a projection whose range is the spherically symmetric functions. In particular, a function f is spherically symmetric if and only if Ãf = f. Moreover, the restriction A of à to l 2 (V, m) is bounded and symmetric. Therefore, A is an orthogonal projection. Definition 2.3 Let the normalized potential q : V [0, ) be given by q = c m. We call a weighted graph (V, b, c, m) weakly spherically symmetric if it contains a vertex x 0 such that κ ± and q are spherically symmetric functions. We call the vertex x 0 in the definition above the root of (V, b, c, m). Remark We will often suppress the dependence on the vertex x 0. Mostly, we will denote weakly spherically symmetric graphs by (V sym, b sym, c sym, m sym ) although b sym, c sym and m sym might not have any obvious symmetries at all. Furthermore, we will denote, as needed, the corresponding curvatures and potential by κ sym ± and qsym and the Laplacian on such a graph by L sym. For a weakly spherically symmetric graph the operator L acts on a spherically symmetric function f by L f(r) = κ + (r)( f (r) f (r + 1)) + κ (r)( f (r) f (r 1)) + q(r) f (r). y S r Moreover, a straightforward calculation yields that κ + (r)m(s r ) = κ (r + 1)m(S r+1 ) for all r N 0. (2.1) Let us give some examples to illustrate the definition of weakly spherically symmetric graphs. Example (a) We call a weighted graph (V, b, c, m) spherically symmetric with respect to x 0 if for each x, y S r (x 0 ), r N 0, there exists a weighted graph automorphism which leaves x 0 invariant and maps x to y. In this case, the weighted graph is weakly spherically symmetric. (b) If the functions k ± := κ ± m, the potential c and the measure m are all spherically symmetric functions, then the weighted graph is weakly spherically symmetric. On the other hand, given that the measure m is spherically symmetric and the graph is weakly spherically symmetric, then k ± and c must be spherically symmetric. Remark The second example shows that there are very little assumptions on the symmetry of the geometry in the weakly spherically symmetric case. The difference is illustrated in Fig. 1. There, standard graphs, that is, with c 0andb : V V {0, 1}, and constant measure are plotted up to the third sphere. The first and second graphs are only weakly spherically symmetric, while the third one is spherically symmetric. However, given the lack of assumptions on connections within a sphere and the structure of connections between the spheres, the freedom in the weakly spherically symmetric case is even much greater than illustrated in the figure. Indeed, we do not have any assumptions on the vertex degree as long as the outer and inner curvatures are constant on each sphere. 123

147 910 M. Keller et al. Fig. 1 The first two standard graphs are only weakly spherically symmetric while the third one is spherically symmetric 3 Heat kernel comparisons 3.1 Notations and definitions Let a weighted graph (V, b, c, m) be given. For the operator L acting on D(L) l 2 (V, m) we know, by the discreteness of the underlying space, that there exists a map p :[0, ) V V R, which we call the heat kernel associated to L, with e tl f (x) = y V p t (x, y) f (y)m(y) for all f l 2 (V, m). Here, e tl is the operator semigroup of L which is defined via the spectral theorem. By direct computation, one sees that p t (x, y) = e tl δ y (x), forx, y V and t 0, where δ y is a l 1 -normalized delta function, i.e., δ y (x) = m(y) 1 if x = y and zero otherwise. Definition 3.1 We say that the heat kernel p of an operator L is spherically symmetric if there is a vertex x 0 such that the averaging operator A and the semigroup e tl commute for all t 0. In particular, in this case, the function p t (x 0, ) is spherically symmetric for each t and, whenever this is the case, we write p sym t (r) = p t (x 0, x) for x S r and r N 0. In order to compare a general weighted graph with a weakly spherically symmetric one, we introduce the following terminology. Definition 3.2 A weighted graph (V, b, c, m) has stronger (respectively, weaker) curvature growth with respect to x 0 V than a weakly spherically symmetric graph (V sym, b sym, c sym, m sym ) with root o if, m(x 0 ) = m sym (o) and if, for all r N 0 and x S r V, κ + (x) κ sym + (r) and κ (x) κ sym (r) (respectively, κ + (x) κ sym + (r) and κ (x) κ sym (r)). Remark The assumption m(x 0 ) = m sym (o) is a normalization condition, necessary for our comparison theorems below, which states that the same amount of heat enters both graphs at the root. This follows since, in general, p 0 (x, y) = m(x) 1 if x = y and 0 otherwise. 123

148 Volume growth, spectrum and stochastic completeness Theorems and remarks There are two main results about heat kernels which are proven in this section. The first one, which is an analogue for Proposition 2.2 in [12] from the manifold setting, is that weakly spherically symmetric graphs can be characterized by the symmetry of the heat kernel. Theorem 1 (Spherical symmetry of heat kernels) A weighted graph (V, b, c, m) is weakly spherically symmetric if and only if the heat kernel is spherically symmetric. Remark It is clear that the heat kernel on a spherically symmetric graph is spherically symmetric. However, the theorem also implies that the heat kernels of all the graphs illustrated in Fig. 1 are spherically symmetric. The second main result of this section is a heat kernel comparison between weakly spherically symmetric graphs and general weighted graphs. These comparisons were originally inspired by [12] and can be found in [56] for the graph Laplacian and trees. See also [51] for related results in the case of the graph Laplacian, regular trees, and heat kernels with a discrete time parameter. Theorem 2 (Heat kernel comparison with weakly spherically symmetric graphs) If a weighted graph (V, b, m) with heat kernel p has stronger (respectively, weaker) curvature growth than a weakly spherically symmetric graph (V sym, b sym, m sym ) with heat kernel p sym, then for x S r (x 0 ) V,r N 0, and t 0, p sym t (r) p t (x 0, x) 3.3 Proofs of Theorems 1 and 2 ( respectively, p sym t (r) p t (x 0, x) ). Westart by developing theideas necessary for theproofof Theorem1. We first characterize the class of weakly spherically symmetric graphs using Ã, the averaging operator, in the following way. Lemma 3.3 Let (V, b, c, m) be a weighted graph and x 0 V. Let à be the averaging operator associated to x 0. Then, the following assertions are equivalent: (i) (V, b, c, m) is weakly spherically symmetric, i.e., κ ± and q are spherically symmetric functions. (ii) L commutes with Ã, i.e., à L f = L à f for all f C(V ). (iii) L 0 commutes with A on C c (V ), i.e., AL 0 g = L 0 Ag for all g C c (V ). Proof The direction (i) (ii) follows by straightforward computation using the formulas κ + (r)m(s r ) = κ (r + 1)m(S r+1 ),see(2.1). As A and L are restrictions of à and L, their matrix elements agree. This gives the direction (ii) (iii). We finally turn to (iii) (i). The function 1 Sr, which is one on S r and zero elsewhere, satisfies κ + (x) + κ (x) + q(x) if x S r L 0 A1 Sr (x) = L 0 1 Sr (x) = κ (x) if x S r±1 0 otherwise and 1 m(s r ) y S r (κ + (y) + κ (y) + q(y))m(y) if x S r AL 0 1 Sr (x) = 1 m(s r±1 ) y S r±1 κ (y)m(y) if x S r±1 0 otherwise. 123

149 912 M. Keller et al. By (iii), these two expressions must be equal which yields that κ ± and q must be spherically symmetric functions which is (i). From Corollary A.8 in Appendix A with H = l 2 (V, m) and D 0 = C c (V ) we can now infer the following statement. Recall that A is the restriction of à to l 2 (V, m) which is an orthogonal projection onto the subspace of spherically symmetric functions. Lemma 3.4 Let (V, b, c, m) be a weighted graph. Then, the following assertions are equivalent: (i) AL 0 f = L 0 A f for all f C c (V ). (ii) A maps D(L) into D(L) and AL f = L Af for all f D(L). (iii) e tl commutes with A for all t 0. Remark A proof of the implication (i) (ii) in Lemma 3.4 could also be based on Proposition A.4 from the appendix and the approximation of the heat kernel p by the restricted kernels p i on H i := l 2 (B i, m i ) discussed below. Proof of Theorem 1 To prove Theorem 1 simply combine Lemmas 3.3 and 3.4. The following construction of the heat kernel, first presented in the continuous setting in [14] and carried over to our set-up in [37], will be crucial. Let x 0 V and B i = B i (x 0 ) denote the corresponding distance balls. By the connectedness of the graph, V = i=0 B i and, as B i B i+1, the balls are an increasing exhaustion sequence of the graph (V, b, c, m). For i N 0,letm i be the restriction of m to B i and consider the restriction L (D) i of the Laplacian L to the finite dimensional space l 2 (B i, m i ) with Dirichlet boundary conditions. This operator can be defined by restricting the form Q to C c (B i ) and taking the closure in l 2 (B i, m i ) with respect to, Q. It turns out, (see [37]), that L (D) i f (x) = L f(x) for f l 2 (B i, m i ) and x B i. This just means that L (D) i = π i Lι i for the canonical injection ι i : l 2 (B i, m i ) l 2 (V, m) acting as extension by zero and π i, the adjoint of ι i. We let p i denote the heat kernel of L (D) i which is extended to V V by zero. That is, { pt i (x, y) = e tl(d) i δ y (x) if x, y B i, t 0 0 otherwise. Here, δ y (x) is the l 1 -normalized delta function as before. We call p i the restricted heat kernels and note that each p i satisfies ( L + dt d ) p i t (x, y) = 0 forall x, y B i, t 0. Furthermore, by Proposition 2.6 in [37] lim i pi t (x, y) = p t(x, y) for all x, y V. Therefore, to prove a property of the heat kernel it often suffices to prove the corresponding property for the reduced heat kernels and then pass to the limit. This is used repeatedly below. In order to prove Theorem 2, we need a version of the minimum principle for the heat equation in our setting. For U V we let U c = V \ U. Lemma 3.5 (Minimum principle for the heat equation) Let (V, b, c, m) be a weighted graph, U V a connected proper subset and u : V [0, T ] R such that u(x, ) is continuously differentiable for every x U. Suppose that 123

150 Volume growth, spectrum and stochastic completeness 913 (a) ( L + dt d )u 0 on U [0, T ], (b) the negative part u := min{u, 0} of u attains its minimum on U [0, T ], (c) u 0 on (U c [0, T ]) (U {0}). Then, u 0 on U [0, T ]. Proof Let (x, t) be the minimum of u on U [0, T ]. Ifu(x, t) 0, then we are done. Furthermore, by (c), we may assume that t > 0. Therefore, suppose that u(x, t) <0where t > 0. As (x, t) is a minimum for u, wehavethat( dt d u)(x, t) = 0ift (0, T ) and ( dt d u)(x, t) 0ift = T.Since(x, t) is also a minimum with respect to x it follows that ( L + dt d ) u(x, t) Lu(x, t) = m(x) 1 b(x, y) (u(x, t) u(y, t)) + c(x) m(x) u(x, t) 0. y V Therefore, by (a), Lu(x, t) = 0sothatu(x, t) = u(y, t) <0forally x. Repeating the argument we eventually reach some y U contradicting (c). We also need the following extension of Lemma 3.10 from [56] which states that the heat kernel on a weakly spherically symmetric graph decays with respect to r. The proof can be carried over directly to our situation so we omit it here. Lemma 3.6 (Heat kernel decay [56, Lemma 3.10]) Let (V, b, m) be a weighted graph, p be the heat kernel and p i be the restricted kernels of B i. Assume that p i t (x 0, ) is a spherically symmetric function for all t 0 with respect to some vertex x 0. Then, given 0 r i, we have for all t > 0 and, in general, for all r N 0 and t 0 pt i (r) >pi t (r + 1) p t (r) p t (r + 1). With these preparations, we can now prove the second main theorem as follows: Proof of Theorem 2 Let p sym,i be the restricted heat kernels of the weakly spherically symmetric graph (V sym, b sym, m sym ) given in the statement of the theorem. On the general weighted graph (V, b, m), we define the functions ρ i t : V R, t 0, by ρt i (x) := psym,i t (r) for x S r (x 0 ), 0 r i and ρt i (x) := 0 otherwise. Under the assumptions of stronger curvature growth and using the heat kernel decay, Lemma 3.6, it follows from the action of L on spherically symmetric functions given after Definition 2.3 that, for all 0 r i and x S r (x 0 ) V, Lρ t i (x) = κ +(x)(p sym,i t κ sym + (r)(psym,i t = L sym p sym,i t (r). (r) p sym,i t (r) p sym,i t (r + 1)) + κ (x)(p sym,i t (r) p sym,i t (r 1)) (r + 1)) + κ sym (r)(psym,i t (r) p sym,i t (r 1)) Hence, ( L + dt d )ρi t (x) ( L sym + dt d )psym,i t (r) = 0. Let p i be the restricted heat kernels of the general weighted graph (V, b, m) so that ( L + dt d )pi t (x 0, ) = 0onB i V and let u(, t) = ρt i ( ) pi t (x 0, ). It follows that ( L + dt d ) u(x, t) 0 123

151 914 M. Keller et al. on B i [0, T ]. By compactness, the negative part of u(x, t) attains its minimum on B i [0, T ]. Furthermore, as ρt i = 0onBi c by definition, we have u(x, t) = 0onBi c and ρ0 i ( ) = pi 0 (x 0, ) 1 which is m(x 0 ) = m 1 sym (o) for x 0 and 0 otherwise. Hence, by the minimum principle, Lemma 3.5, we get that p sym,i t (r) = ρt i (x) pi t (x 0, x) for x B i. The desired result now follows by letting i. The inequality in the case of weaker curvature growth is proven analogously. 4 Spectral estimates In this section we give an estimate for the bottom of the spectrum and a criterion for discreteness of the spectrum in the weakly spherically symmetric case. We then use the heat kernel comparisons obtained above to give estimates on the bottom of the spectrum for general weighted graphs. 4.1 Notations and definitions Let a weighted graph (V, b, m) be given. Let σ(l) denote the spectrum of L and λ 0 := λ 0 (L) := inf σ(l). We call λ 0 the bottom of the spectrum or the ground state energy. The ground state energy can be obtained by the Rayleigh-Ritz quotient (see, for instance, [47]) as follows: λ 0 = Q( f, f ) inf f D(Q) f, f Lf, f = inf f C c (V ) f, f where the last equality follows since C c (V ) is dense in D(Q) with respect to, Q =, + Q(, ) as discussed in Sect. 2. Furthermore, the spectrum of an operator may be decomposed as a disjoint union as follows: σ(l) = σ disc (L) σ ess (L) where σ disc (L) denotes the discrete spectrum of L, defined as the set of isolated eigenvalues of finite multiplicity, and σ ess (L) denotes the essential spectrum,givenbyσ ess (L) = σ(l) \ σ disc (L). We will use the notation λ ess 0 (L) to denote the bottom of the essential spectrum of L. Fix a vertex x 0 and let S r = S r (x 0 ) and B r = B r (x 0 ). Definition 4.1 For f : V R and r N 0,wedefinetheweighted volume of a ball by V f (r) = x B r f (x)m(x). In particular, V 1 (r) = m(b r ). Moreover, for r N 0,welet B(r) = x S r κ + (x)m(x) be the measure of the boundary of a ball which is the weight of the edges leaving the ball. Note that, in the weakly spherically symmetric case, B(r) = κ sym + (r)msym (S r ). 123

152 Volume growth, spectrum and stochastic completeness Theorems and remarks The first main result of this section is an estimate on the bottom of the spectrum and a criterion for discreteness of the spectrum in terms of volume and boundary growth for weakly spherically symmetric graphs. Theorem 3 (Volume and spectrum) Let (V sym, b sym, m sym ) be a weakly spherically symmetric graph. If then r=0 V 1 (r) B(r) = a <, λ 0 (L sym ) 1 a and σ(lsym ) = σ disc (L sym ). The second main result of this section is a comparison of the bottom of the spectrum of a general weighted graph and a weakly spherically symmetric one. Theorem 4 (Spectral comparison) If a weighted graph (V, b, m) has stronger (respectively, weaker) curvature growth than a weakly spherically symmetric graph (V sym, b sym, m sym ), then λ 0 (L) λ 0 (L sym ) (respectively, λ 0 (L) λ 0 (L sym )). If (V sym, b sym, m sym ) satisfies V 1 (r) r=0 B(r) growth, then = a < and (V, b, m) has stronger curvature λ 0 (L) 1 a and σ(l) = σ disc(l). Remarks Let us discuss these results in light of the present literature: (a) In Sect. 6, we give examples of standard graphs with m 1 of polynomial volume growth satisfying the summability criterion above. For such graphs, it follows that has positive bottom of the spectrum as well as discrete spectrum. This stands in clear contrast to the celebrated theorem of Brooks for Riemannian manifolds [9] and results for [22,32] since, for these, subexponential volume growth always implies that the bottom of the essential spectrum is zero. (b) For statements analogous to Theorem 3 for the Laplacian on spherically symmetric Riemannian manifolds see [2,29]. (c) There are many examples of estimates for the bottom of the spectrum for the graph Laplacian and normalized graph Laplacian, see, for example, [7,15 19,40,44,52, 53]. In particular, from the analogue of the Cheeger inequality found in [15] it follows that λ 0 ( ) = 0if B(r) V 1 (r) 0asr.OurTheorem3 complements these results by giving a lower bound for λ 0 ( ) in the case of unbounded vertex degree. For, our result is not applicable as m(x) = deg(x) implies that V 1 (r) B(r). (d) Discreteness of the spectrum of the graph Laplacians was studied in [23,36,52,56]. In our context, a characterization for weighted graphs with positive Cheeger s constant at infinity to have discrete spectrum was recently given in [38]. In Sect. 6,wediscusshow our results complement these and are not implied by any of them, see Corollaries 6.6 and

153 916 M. Keller et al. (e) With regards to Theorem 4, itwasshownin[10] that the bottom of the spectrum of the graph Laplacian on a k-regular graph is smaller than that on a k-regular tree. This was generalized from k-regular to arbitrary graphs with degree bounded by k in [52], which also contains a corresponding lower bound. Analogous statements for were proven by different means in [58]. 4.3 Proofs of Theorems 3 and 4 The proofs of the first statement in Theorem 3 and the second statement in Theorem 4 are based on the following characterization for the bottom of the spectrum, which is sometimes referred to as a Allegretto Piepenbrink type of theorem. We refer to [27,Theorem3.1]for a proof and further discussion of earlier results of this kind. Proposition 4.2 (Characterization of the bottom of the spectrum, [27, Theorem 3.1]) Let (V, b, c, m) be a weighted graph. For α R the following statements are equivalent: (i) There exists a non-trivial v : V [0, ) suchthat ( L + α)v 0. (ii) There exists v : V (0, ) such that ( L + α)v = 0. (iii) α λ 0 (L). Therefore, to prove a lower bound on the bottom of the spectrum, it is sufficient to demonstrate a positive (super-)solution to the difference equation above. In the weakly spherically symmetric case, we will look for spherically symmetric solutions. We now state and prove the following lemma which generalizes a result of [57] and gives the existence of solutions for any initial condition. Note that we allow for a non-negative potential as we will also use this statement in the next section on stochastic completeness. Lemma 4.3 (Recursion formula for solutions) Let (V sym, b sym, c sym, m sym ) be a weakly spherically symmetric graph and α R. A spherically symmetric function v is a solution to ( L sym + α)v(r) = 0 if and only if v(r + 1) v(r) = 1 B(r) r C q+α ( j)v( j) where C q+α ( j) = (q sym ( j) + α)m sym (S j ). In particular, v is uniquely determined by the choice ofv(0). Consequently, if v(0) > 0 and α > 0, then v is a strictly positive, monotonously increasing solution. Proof The proof is by induction. For r = 0, ( L sym + α)v(0) = 0gives ( L sym + α)v(0) = κ sym + (0) (v(0) v(1)) + ( q sym (0) + α ) v(0) = 0 which yields the assertion. Assume now that the recursion formula holds for r 1 where r 1. Then, ( L sym + α)v(r) = 0 reads as κ sym + (r) (v(r) v(r + 1)) + κsym (r) (v(r) v(r 1)) + ( q sym (r) + α ) v(r) = j=0

154 Volume growth, spectrum and stochastic completeness 917 Therefore, v(r + 1) v(r) = κsym (r) κ sym + (r) (v(r) v(r 1)) + 1 ( κ sym q sym + (r) (r) + α ) v(r) = κsym (r) κ sym + (r) + 1 r 1 κ sym + (r C q+α ( j)v( j) 1)msym (S r 1 ) 1 ( κ sym q sym + (r) (r) + α ) v(r) = 1 B(r) r C q+α ( j)v( j) j=0 by κ sym + (r 1)msym (S r 1 ) = κ sym (r)msym (S r ) as noted in (2.1). Whenever α>0, the right hand side of the recursion formula is positive from the assumption that v(0) >0 which gives the monotonicity statement. In order to prove the statements of Theorems 3 and 4 concerning the essential spectrum, we need to restrict our operator L to the complements of balls. For i N 0,letBi c = V \ B i and mi c be the restriction of m to Bc i.werestricttheformq to C c(bi c ) and take the closure in l 2 (Bi c, mc i ) with respect to, Q. By standard theory, we obtain an operator on l 2 (Bi c, mc i ) which we call the restriction of L with Dirichlet boundary conditions and which we denote by L (D) i. Note that, in contrast to the previous section, these operators are now defined on the complement of balls and hence on infinite dimensional spaces. Lemma 4.4 (Existence of strictly positive solutions) Let (V sym, b sym, m sym ) be a weakly spherically symmetric graph. Suppose that a = V 1 (r) r=0 B(r) <. Then, there exists a strictly positive, strictly monotone decreasing spherically symmetric solution v on V sym to ( L sym a 1 )v = 0 which satisfies j=0 v(r + 1) 1 1 a r j=0 V 1 ( j) B( j) for all r N 0. Moreover, for all i N 0, there exists a strictly positive, strictly monotone decreasing function v i on Bi c solving ( L (D) i a 1 i )v i = κ sym (i + 1)1 S i+1 which satisfies v i (r + 1) 1 1 a i r j=i+1 V i 1 ( j) B( j) for all r i + 1, where 1 Si+1 (x) is 1 for x S i+1 and 0 otherwise, a i = j=i+1 V 1 ( j) B( j) and V i 1 ( j) = m sym (B j \ B i ). Proof For α = 1 a and csym 0 the recursion formula of Lemma 4.3 reads as v(r + 1) v(r) = 1 a B(r) r C 1 ( j)v( j), where C 1 ( j) = m sym (S j ). Hence, there exists a solution v of the equation for v(0) >0. In order to prove our assertion, we show by induction that for all r N 0, j=0 123

155 918 M. Keller et al. (i) v(r + 1) <v(r) (ii) v(r + 1) (iii) v(r + 1) >0. ( 1 1 a ) rj=0 V 1 ( j) B( j) v(0) For r = 0, we get from the recursion formula above that v(1) v(0) = 1 a B(0) msym (0)v(0) <0 which gives (i). Furthermore, ( v(1) = 1 V ) 1(0) v(0) a B(0) which gives (ii) and (iii) follows by the choice of a. Now, suppose that (i), (ii), and (iii) hold for r > 0. Then, since v(j) > 0for j = 0, 1,...,r, the recursion formula above yields that v(r + 1) v(r) <0 which gives (i). Furthermore, 1 r v(r + 1) = v(r) C 1 ( j)v( j) >v(r) V 1(r) a B(r) a B(r) v(0) 1 1 a r 1 j=0 j=0 V 1 ( j) v(0) B( j) V 1(r) a B(r) v(0) = 1 1 a r V 1 ( j) v(0) B( j) which yields (ii) and (iii) follows by the choice of a. For the second statement, we define the function v i on B c i by v i (i + 1) = 1and v i (r) = v i (r 1) 1 a i B(r 1) r 1 j=i+1 j=0 C 1 ( j)v i ( j) for r > i + 1. By a direct calculation one checks that (L (D) i a 1 i )v i (i + 1) = κ (i + 1) 0 and, as in the proof of Lemma 4.3, that(l (D) i a 1 i )v i (r) = 0forr > i + 1. Now, by the same arguments as above, one shows that v i is strictly monotone decreasing, satisfies v i (r) 1 1 a i r 1 j=i+1 V i 1 ( j) B( j) and is strictly positive. Proof of Theorem 3 By Lemma 4.4, there is a strictly positive solution to ( L sym a 1 )v = 0 where a = V 1 (r) r=0 B(r).Thisprovesthatλ 0(L sym ) a 1 by the characterization of the bottom of the spectrum, Proposition 4.2. We now show that σ(l sym ) = σ disc (L sym ). By standard theory (see, for example, Proposition 18 in [38]) if follows that λ ess 0 (Lsym ) = lim λ 0(L (D) i i ). By Proposition 4.2 and the second part of Lemma 4.4, wehavethatλ 0 (L (D) i ) a 1 i. Since a i 0asi, it follows that λ 0 (L (D) i ) as i so that σ ess (L sym ) =,thatis, σ(l sym ) = σ disc (L sym ). In order to prove the spectral comparison, Theorem 4, we need an analogue of the wellknown theorem of Li which links large time heat kernel behavior and the ground state energy [11,41,46]. It was recently proven for our setting in [28,39]. 123

156 Volume growth, spectrum and stochastic completeness 919 Proposition 4.5 (Heat kernel convergence to ground state energy [28,39]) Let (V, b, c, m) be a weighted graph. For all vertices x and y, ln p t (x, y) lim = λ 0 (L). t t Theorem 4 now follows directly: Proof of Theorem 4 The first statement follows by combining Theorem 2 and Proposition 4.5. For the second statement, we first derive λ 0 (L) a 1 from the first statement and Theorem 3. Now,letv i be the strictly positive monotone decreasing functions which solve ( L (D) a 1 i )v i = κ sym (i + 1)1 S i+1 on Bi c V sym with a i = V 1 ( j) j=i+1 B( j) which exist according to Lemma 4.4 for all i 0. We choose v i to be normalized by letting v i (i +1) = 1. We define w i on Bi c V via w i (x) = v i (r) for x S r, r i + 1. Clearly, by the stronger curvature growth, ( L (D) i a 1 i )w i (x) ( L sym,(d) i a 1 i )v i (r) for x S r, r > i + 1. On the other hand, for r = i + 1andx S r we have, again by the stronger curvature growth, that ( L (D) i 1 a i )w i (x) ( L sym,(d) i 1 a i )v i (i + 1) + (κ (x) κ sym (i + 1)) = κ (x) 0 since v i (i + 1) = 1. Hence, ( L (D) i a 1 i )w i 0 with w i strictly positive and we conclude that λ 0 (L (D) i ) a 1 i by Proposition 4.2. Asa i 0, we get, as in the proof of Theorem 3, that σ ess (L) = and, therefore, σ(l) = σ disc (L). 5 Stochastic completeness 5.1 Notations and definitions The study of the uniqueness of bounded solutions for the heat equation has a long history in both the discrete, see, for example, [20,48], and the continuous, see [25] and references therein, settings. In recent years, there has been interest in finding geometric conditions for infinite graphs implying this uniqueness, see, for example, [16,19,26,33,37,38,54 57]. In the general setting of [37] it is shown that this uniqueness is equivalent to several other properties as we discuss below. Let a weighted graph (V, b, c, m) be given. We let u 0 : V R be bounded and call u : V [0, ) R a solution of the heat equation with initial condition u 0 if, for all x V, u(x, ) is continuous on [0, ), differentiable on (0, ) and satisfies { ( L + dt d ) u(x, t) = 0 for x V, t > 0, u(x, 0) = u 0 (x) for x V. The question of the uniqueness of bounded solutions for the heat equation on (V, b, c, m) is then reduced to having u 0 be the only bounded solution for the heat equation with initial condition u 0 0. In order to study this question, the following function which was introduced in [37]turns out to be essential. Let M t (x) = e tl 1(x) + t 0 e sl q(x)ds, where q = m c is the normalized potential, 1 denotes the function whose value is 1 on all vertices and e tl is the operator semigroup extended to the space of bounded functions on V. 123

157 920 M. Keller et al. The function e sl q is defined as the pointwise limit along the net of functions g C c (V ) such that 0 g q, where the net is considered with respect to the natural ordering g h whenever g h. As shown in [37], this limit always exists and 0 M t (x) 1forallx V. The function M t consists of two parts which can be interpreted as follows: the first term, e tl 1(x), is the heat which is still in the graph at time t. The integral denotes the heat which was killed by the potential up to time t. Thus, 1 M t can be interpreted as the heat which is transported to the boundary of the graph. In this setting, Theorem 1 of [37] (see also Proposition 28 of [38]) states the following: Proposition 5.1 (Characterization of stochastic completeness [37, Theorem 1]) Let (V, b, c, m) be a weighted graph. The following statements are equivalent: (i) There exists v : V [0, ) non-zero, bounded such that ( L + α)v 0 for some (equivalently, all) α>0. (ii) There exists v : V (0, ) bounded such that ( L + α)v = 0 for some (equivalently, all) α>0. (iii) M t (x) <1 for some (equivalently, all) x V and t > 0. (iv) There exists a non-trivial, bounded solution to the heat equation with initial condition u 0 0. Such weighted graphs are called stochastically incomplete at infinity. Otherwise, a weighted graph is called stochastically complete at infinity. This extends the usual notion of stochastic completeness for the Laplacian to the case where a potential is present. Clearly, as already discussed in [37], when the potential is zero, a stochastically complete weighted graph is also stochastically complete at infinity. In order to formulate our stochastic completeness comparison theorems, we need to compare the potentials of two weighted graphs, continuing Definition 3.2. Definition 5.2 We say that a weighted graph (V, b, c, m) has stronger (respectively, weaker) potential with respect to x 0 V than a weakly spherically symmetric graph (V sym, b sym, c sym, m sym ) if, for all x S r (x 0 ) V and r N 0, q(x) q sym ( (r) respectively, q(x) q sym (r) ). 5.2 Theorems and remarks It is desirable to have conditions which imply stochastic completeness or incompleteness at infinity and this is our goal. We start with a characterization of stochastic completeness at infinity for weakly spherically symmetric graphs whose proof will be given at the end of the section. It generalizes a result for the graph Laplacian on spherically symmetric graphs found in [57]. Theorem 5 (Geometric characterization of stochastic completeness) A weakly spherically symmetric graph (V sym, b sym, c sym, m sym ) is stochastically complete at infinity if and only if where r=0 V q+1 (r) B(r) = V q+1 (r) = x B r (q sym (x) + 1)m sym (x) and B(r) = κ sym + (r)msym (S r ). 123

158 Volume growth, spectrum and stochastic completeness 921 Combining this with Theorem 3 we get an immediate corollary, which is an analogue to theorems found in [2,29] for the Laplacian on a Riemannian manifold. The proof is given right after the proof of Theorem 5. Corollary 5.3 If a weakly spherically symmetric graph (V sym, b sym, c sym, m sym ) is stochastically incomplete at infinity, then λ 0 (L sym )>0 and σ(l sym ) = σ disc (L sym ). Remarks (a) The converse statements do not hold. For example, in the standard case, both and on regular trees of degree greater than 2 have positive bottom of the spectrum but govern stochastically complete processes. Furthermore, as shown in [36,52], for a tree one has σ( ) = σ disc ( ) whenever the vertex degree goes to infinity along any sequence of vertices which eventually leaves every finite set, while stochastic incompleteness requires that the vertex degree goes to infinity at a certain rate as shown in [56](seealso Sect. 6). (b) The statements of the corollary do not hold for general weighted graphs. This can be seen from stability results for stochastic incompleteness at infinity proven in [33,37,56]which state that attaching any graph to a graph which is stochastically incomplete at infinity at a single vertex does not change the stochastic incompleteness. Therefore, starting with a stochastically incomplete spherically symmetric tree, attachment of a single path to infinity can drive the bottom of the spectrum down to zero and add essential spectrum (as follows by general principles) without effecting the stochastic incompleteness. (c) The two statements of the corollary are not completely independent: if a Laplacian on a graph has purely discrete spectrum and the constant function 1 does not belong to l 2 (V, m) or c 0, then the lowest eigenvalue cannot be zero. Our second main result of this section is a comparison theorem for stochastic completeness in the spirit of [34]. Theorem 6 (Stochastic completeness at infinity comparison) If a weighted graph (V, b, c, m) has stronger curvature growth and weaker potential (respectively, weaker curvature growth and stronger potential) than a weakly spherically symmetric graph (V sym, b sym, c sym, m sym ) which is stochastically incomplete (respectively, complete) at infinity, then (V, b, c, m) is stochastically incomplete (respectively, complete) at infinity. Remarks (a) Note that a stronger potential can make a stochastically incomplete graph stochastically complete at infinity (compare with Theorem 2 in [37]). This is due to the definition of M t. Specifically, the potential kills heat in the graph and, as such, prevents it from being transported to infinity. (b) For the results above to hold, it suffices that the comparisons hold outside of a finite set. This is due to the fact that stochastic (in) completeness is stable under finite dimensional perturbations, compare [33,37,56]. 5.3 Proofs of Theorems 5 and 6 We begin with an observation concerning the solutions to the difference equation on weakly spherically symmetric graphs which we have encountered before. Lemma 5.4 (Boundedness of spherically symmetric solutions) Let a weakly spherically symmetric graph (V sym, b sym, c sym, m sym ) be given. Let α>0 and v : V sym (0, ) be 123

159 922 M. Keller et al. a spherically symmetric function such that ( L sym + α)v = 0. Then, v is unbounded if and only if r=0 V q+1 (r) B(r) =. Proof We will use the obvious fact that V q+1 (r) r=0 B(r) = if and only if V q+α (r) r=0 B(r) = for some (equivalently, all) α>0. By the recursion formula of Lemma 4.3, the function v satisfies v(r + 1) v(r) = 1 B(r) r C q+α (i)v(i) where C q+α (r) = (q sym (r) + α)m sym (S r ) and consequently is monotonously increasing as v(0) >0. Hence, it satisfies v(r + 1) v(r) V q+α(r) B(r) v(0) where V q+α (r) = x B r (q sym (x) + α)m sym (x) = r i=0 C q+α (i). Therefore, if r=0 V q+1 (r) B(r) =,thenv(r) = r 1 i=0 (v(i + 1) v(i)) as r. Hence, v is unbounded in this case. On the other hand, the recursion formula and monotonicity imply that v(r + 1) ( 1 + V ) q+α(r) v(r) B(r) i=0 r i=0 ( 1 + V ) q+α(i) v(0). B(i) Therefore, if V q+1 (r) r=0 B(r) <, then ( ) r=0 1 + V q+α(r) B(r) < so that v is bounded. We combine the lemma above with the characterizations of Proposition 5.1 in order to prove stochastic incompleteness at infinity of weakly spherically symmetric graphs. For the other direction, we will need the following criterion for stochastic completeness at infinity which is an analogue for a criterion of Has minskiĭ from the continuous setting [30]. Recently, Huang [33] has proven a slightly stronger version in the case c 0. For v : V R, we write v(x) as x whenever for every C 0 there is a finite set K such that v V \K C. Proposition 5.5 (Condition for stochastic completeness) If on a weighted graph (V, b, c, m) there exists v such that ( L + α)v 0 and v(x) as x, then (V, b, c, m) is stochastically complete at infinity. Proof Suppose there is a function 0 w 1 that solves ( L + α)w = 0. For given C > 0 let K V be a finite set such that v V \K C. Thenu = v Cw satisfies ( L + α)u 0 on V and u 0onV \ K.AsK is finite, the negative part of u attains its minimum on K. Therefore, by a minimum principle, see [37, Theorem 8], u 0onK. Hence, v Cw for all C > 0. This implies that w 0. By Proposition 5.1, stochastic completeness at infinity follows. 123

160 Volume growth, spectrum and stochastic completeness 923 Proof of Theorem 5 By Lemma 4.3, there is always a strictly positive, spherically symmetric solution to ( L sym + α)v = 0forα>0. By Lemma 5.4, this solution is bounded if and only if the sum in the statement of Theorem 5 converges. In the case of convergence, we conclude stochastic incompleteness at infinity by Proposition 5.1. On the other hand, if the sum diverges, the solution is unbounded and satisfies the assumptions of Proposition 5.5 (by the spherical symmetry). Hence, the graph is stochastically complete at infinity. Proof of Corollary 5.3 If the graph (V sym, b sym, c sym, m sym ) is stochastically incomplete at infinity, then the sum of Theorem 5 converges. As a consequence, the sum of Theorem 3 converges which implies positive bottom of the spectrum and discreteness of the spectrum of the graph (V sym, b sym, m sym ). Now, a non-negative potential only lifts the bottom of the spectrum (as can be seen from the Rayleigh-Ritz quotient) which gives the statement. Proof of Theorem 6 Let (V sym, b sym, c sym, m sym ) be a weakly spherically symmetric graph. Let v be the spherically symmetric solution with v(0) = 1 given by Lemma 4.3 for α>0. Then, v is strictly positive and monotonously increasing. Define w : V (0, ) on the weighted graph (V, b, c, m) by w(x) = v(r) for x S r (x 0 ) V. First, assume the stochastic incompleteness at infinity of (V sym, b sym, c sym, m sym ). Then, by Lemma 5.4 combined with Theorem 5, the function v, and thus w, is bounded. Under the assumptions that (V, b, c, m) has stronger curvature growth and weaker potential than (V sym, b sym, c sym, m sym ) and, as v is monotonously increasing by Lemma 4.3, the function w satisfies for x S r, r N, ( L + α ) w(x) = κ + (x) (v(r) v(r + 1)) + κ (x) (v(r) v(r 1)) + (q(x) + α) v(r) ( L sym + α ) v(r) = 0. Hence, the graph (V, b, c, m) is stochastically incomplete at infinity, by Proposition 5.1. Assume now that (V sym, b sym, c sym, m sym ) is stochastically complete at infinity. Then, again by Proposition 5.1, the function v and, thus w, is unbounded. As above, under the assumptions of weaker curvature growth and stronger potential, one checks that ( L + α ) w(x) ( L sym + α ) v(r) = 0 for all x S r V, r N 0.Sincew is unbounded, w(x) as x, therefore, the weighted graph (V, b, c, m) is stochastically complete at infinity by the condition for stochastic completeness, Proposition Applications to graph Laplacians In this section we discuss the results of this paper for standard graphs. Thus, we consider the situation b : V V {0, 1} and c 0. Chosing m 1andm deg we obtain the two graph Laplacians, that is, on l 2 (V ) = l 2 (V, 1) and on l 2 (V, deg). Note that, in the setting for, the curvatures κ ± : V N denote the number of edges connecting a vertex x, which lies in the sphere of radius r = d(x, x 0 ), to vertices in the spheres of radius r ± 1. Therefore, we can think of κ and κ + as the inner and outer vertex degrees, respectively. On the other hand, for, the curvatures κ ± : V Q are inner and outer vertex degree divided by the degree. In this situation, we have the following corollaries of Theorem 1 for the heat kernels. Corollary 6.1 The operator has a spherically symmetric heat kernel if and only if the inner and outer vertex degrees are spherically symmetric. 123

161 924 M. Keller et al. Fig. 2 A tree and an antitree Corollary 6.2 The operator has a spherically symmetric heat kernel if and only if the ratio of inner and outer vertex degrees and the vertex degree are spherically symmetric. Note that, in order to have a spherically symmetric heat kernel for, even less symmetry than the graphs of Fig. 1 possess is needed. Next, we fix m 1 to focus on and give examples of weakly spherically symmetric graphs which satisfy our summability condition. We start with the case of spherically symmetric trees. Here, κ (r) = 1 and, if V (r) := V 1 (r), the summability criterion of Theorems 3 and 5 concerns the convergence or divergence of r=0 V (r) B(r) = r=0 B r S r+1 which can be seen to be equivalent to the convergence or divergence of r=0 1 κ + (r),see[57]. Using this, it follows easily, as was already shown in [56], that the threshold for the volume growth for stochastic completeness of spherically symmetric trees lies at r! e r log r.this already stands in contrast to the result of Grigor yan which puts the threshold for stochastic completeness of manifolds at e r 2 [25]. However, this is still in line with the result of Brooks which yields that subexponential growth implies absence of a spectral gap for manifolds [9]. We now come to the family of spherically symmetric graphs, here called antitrees, which yield our surprising examples. Definition 6.3 A standard weakly spherically symmetric graph with m 1 is called an antitree if κ + (r) = S r+1 for all r N 0. As opposed to trees, which are connected graphs with as few connections as possible between spheres, antitrees have the maximal number of connections. The contrast is illustrated in Fig. 2. Such graphs have already been used as examples in [17,54,57]. In the case of antitrees, the summability criterion concerns the convergence or divergence of r=0 V (r) B(r) = r=0 B r S r S r+1. For functions f, g : N 0 R we write f g provided that there exist constants c and C such that cg(r) f (r) Cg(r) for all r N 0. We can then relate the summability criterion above to volume growth through the following lemma whose proof is immediate. 123

162 Volume growth, spectrum and stochastic completeness 925 Lemma 6.4 (Volume growth of antitrees) Let β>0. An antitree with κ + (r) r β for r N 0 satisfies V (r) r β+1. Furthermore, if β>2 (respectively, β 2), then ( V (r) ) B(r) < V (r) respectively, B(r) =. r=0 Combining this lemma with Theorem 5 immediately gives the following corollary. Corollary 6.5 (Polynomial growth and stochastic incompleteness) Let β>0. An antitree with κ + (r) r β satisfies V (r) r β+1 and is stochastically complete if and only if β 2. This phenomenon was already observed in [57] and it gives an even sharper contrast to Grigor yan s threshold for manifolds as the volume growth does not even have to be exponential for the antitree to become stochastically incomplete. Furthermore, in [26, Theorem 1.4], it is shown that any graph whose volume growth is less than cubic is stochastically complete. Thus, our examples are, in some sense, the stochastically incomplete graphs with the smallest volume growth. It should also be mentioned that it was recently shown by Huang [33], using ideas found in [1], that the condition V (r) r=0 B(r) = which implies stochastic completeness of weakly spherically symmetric graphs does not imply stochastic completeness for general graphs. Let us now turn to a discussion of the spectral consequences. Combining the volume growth of antitrees, Lemma 6.4, with Theorem 3 gives the following immediate corollary. Corollary 6.6 (Polynomial growth and positive bottom of the spectrum) Let β > 2. An antitree with κ + (r) r β satisfies V (r) r β+1 with r=0 λ 0 ( ) > 0 and σ( ) = σ disc ( ). This corollary shows that, for, there are no direct analogues to Brook s theorem which states, in particular, that subexponential volume growth of manifolds implies that the bottom of the essential spectrum is zero. On the other hand, there is a result by Fujiwara [22] (which was later generalized in [32]) for the normalized Laplacian acting on l 2 (V, deg), which states that λ ess 0 ( ) 1 2e μ e μ. 1 Here, the exponential volume growth is given by μ = lim sup r r log V 1 (r) where V 1 (r) = x B r deg(x). Therefore, if a graph has subexponential volume growth with respect to the measure deg, it follows that λ 0 ( ) λ ess 0 ( ) = 0. Let us also discuss how our Theorem 3 complements some of the results found in [36,38]. There it is shown that, for graphs with positive Cheeger constant at infinity, α > 0, rapid branching is equivalent to discreteness of the spectrum of the Laplacian. The constant α is defined as the limit over the net of finite sets K of the quantities W α K = inf m(w ) where W is the number of edges leaving W and the infimum is taken over all finite sets W V \ K,see[23,36,38]. Now, subexponential volume growth, that is, μ = 0 implies α = 0by1 1 α 2 λess 0 ( ), shown in [23], combined with the estimate for λ ess 0 ( ) given above. Corollary 6.6 gives examples of graphs with μ = 0 and thus α = 0butfor which has no essential spectrum. 123

163 926 M. Keller et al. Another interesting consequence of Theorem 2 and Theorem 4 for is the following: Corollary 6.7 Let G and G be the graph Laplacians of two weakly spherically symmetric graphs which have the same curvature growth. Then, λ 0 ( G ) = λ 0 ( G ). This means, in particular, that the Laplacians on all graphs in Fig. 1 have the same bottom of the spectrum. Note that this is not at all true for the normalized graph Laplacian as it operates on l 2 (V, m) = l 2 (V, deg) and the presence of edges connecting vertices on the same sphere clearly effects the degree measure. To illustrate this contrast, note that on a k-regular tree T k the bottom of the spectrum is known to be λ 0 ( Tk ) = 1 k (k 2 k 1) for all k. On the other hand, if one connects all vertices in each sphere one obtains a graph G k such that λ 0 ( Gk ) = 0 as shown in [36, Theorem 6]. However, from Corollary 6.7 above, λ 0 ( Tk ) = λ 0 ( Gk ) = k 2 k 1which is also new compared to [36, Theorem 6], where only λ 0 ( Gk ) k is shown. In particular, we have another example where the ground state energies of and differ. Acknowledgments The authors are grateful to Józef Dodziuk for his continued support. MK and RW would like to thank the Group of Mathematical Physics of the University of Lisbon for their generous backing while parts of this work were completed. In particular, RW extends his gratitude to Pedro Freitas and Jean-Claude Zambrini for their encouragement and assistance. RW gratefully acknowledges financial support of the FCT in the forms of grant SFRH/BPD/45419/2008 and project PTDC/MAT/101007/2008. Appendix A: Reducing subspaces and commuting operators We study symmetries of selfadjoint operators. These symmetries are given in terms of bounded operators commuting with the selfadjoint operator in question. We present a general characterization in Theorem A.1. With Lemma A.5, we then turn to the question of how symmetries of a symmetric non-negative operator carry over to its Friedrichs extension. Finally, we specialize to the situation in which the bounded operator is a projection onto a closed subspace. The main result of this appendix, Corollary A.8, characterizes when a selfadjoint operator commutes with such a projection. While these results are certainly known in one form or another, we have not found all of them in the literature in the form discussed below. In the main body of the paper they will be applied to Laplacians on graphs. However, they are general enough to be applied to Laplace-Beltrami operators on manifolds as well. A subspace U of a Hilbert space is said to be invariant under the bounded operator A if A maps U into U. Theorem A.1 Let L be a selfadjoint non-negative operator on the Hilbert space H and A a bounded operator on H. Then, the following assertions are equivalent: (i) D(L) is invariant under A and L Ax = ALx for all x D(L). (ii) D(L 1/2 ) is invariant under A and L 1/2 Ax = AL 1/2 x for all x D(L 1/2 ). (iii) 1 [0,t] (L)A = A1 [0,t] (L) for all t 0. (iv) e tl A = Ae tl for all t 0. (v) (L + α) 1 A = A(L + α) 1 for all α>0. (vi) g(l)a = Ag(L) for all bounded measurable g :[0, ) C. Proof This is essentially standard. We sketch a proof for the convenience of the reader. We first show that (iii), (iv), (v) and (vi) are all equivalent: (iii) (iv): This follows by a simple approximation argument. 123

164 Volume growth, spectrum and stochastic completeness 927 (iv) (v): This follows immediately from (L + α) 1 = 0 e tα e tl dt (which, in turn, is a direct consequence of the spectral calculus). (v) (vi): The assumption (v) together with a Stone/Weierstrass-type argument shows that g(l)a = Ag(L) for all continuous g :[0, ) C with g(x) 0forx.Now, it is not hard to see that the set { f :[0, ) C f measurable and bounded with f (L)A = Af(L)} is closed under pointwise convergence of uniformly bounded sequences. This gives the desired statement (vi). (vi) (iii): This is obvious. We now show (ii) (i) (v) and (vi) (ii). (ii) (i): This is clear as L = L 1/2 L 1/2. (i) (v): Obviously, (i) implies A(L + α)x = (L + α)ax for all α R and x D(L). As (L + α) is injective for α>0 we infer for all such α that (L + α) 1 A = A(L + α) 1. (vi) (ii): For every natural number n the operator L 1/2 1 [0,n] (L) = (id 1/2 1 [0,n] )(L) is a bounded operator commuting with A by (vi). Let x D(L 1/2 ) be given and set x n := 1 [0,n] (L)x. Then, x n belongs to D(L 1/2 ). Moreover, as 1 [0,n] (L) is a projection, we obtain by (vi) that Ax n = A1 [0,n] (L)x = 1 [0,n] (L)A1 [0,n] (L)x = 1 [0,n] (L)Ax n. In particular, Ax n belongs to D(L 1/2 ) as well. This gives, by (vi) again, that L 1/2 Ax n = L 1/2 1 [0,n] (L)Ax n = AL 1/2 1 [0,n] (L)x. As x belongs to D(L 1/2 ),weinferthatl 1/2 1 [0,n] (L)x converges to L 1/2 x. Moreover, Ax n obviously converges to Ax.AsL 1/2 is closed, we obtain that Ax belongs to D(L 1/2 ) as well and L 1/2 Ax = L 1/2 Ax holds. Remark (a) The method to prove (v) (vi) can be strengthened as follows: Let L be a selfadjoint operator with spectrum.letb( ) be the algebra of all bounded measurable functions on. A sequence ( f n ) in B( ) is said to converge to f B( ) in the sense of ( )ifthe( f n ) are uniformly bounded and converge pointwise to f.letf be a subset of B such that f (L)A = Af(L) holds for all f F. If the smallest subalgebra of B which contains F and is closed under convergence with respect to ( ) isb, then g(l)a = Ag(L) for all g B. (b) If L is an arbitrary selfadjoint operator then the equivalence of (i), (iii) and (vi) is still true and the semigroup in (iv) can be replaced by the unitary group and the resolvents in (v) can be replaced by resolvents for α C \ R (as can easily be seen using (a) of this remark). Definition A.2 Let L be a selfadjoint non-negative operator on a Hilbert space H and A a bounded operator on H. Then, A is said to commute with L if one of the equivalent statements of the theorem holds. Corollary A.3 Let L be a selfadjoint non-negative operator on a Hilbert space H and A a bounded operator on H. Then, A commutes with L if and only if its adjoint A commutes with L. Proof Take adjoints in (iii) of the previous theorem. 123

165 928 M. Keller et al. A simple situation in which the previous theorem can be applied is given next. Proposition A.4 Let L be a selfadjoint non-negative operator on the Hilbert space H and let A be a bounded operator on H. Let, for each natural number n, a closed subspace H n of H be given with AH n H n and n H n = H. If, for each n, there exists a selfadjoint non-negative operator L n from H n to H n with AL n = L n A and (L n+k + α) 1 x (L + α) 1 x, k, for all natural numbers n, x H n and α>0, thenal = L A holds. A corresponding statement holds with resolvents replaced by the semigroup. Proof By assumption we have A(L + α) 1 x = (L + α) 1 Ax for all x from the dense set n H n. By boundedness of the respective operators we infer A(L + α) 1 = (L + α) 1 A and the statement follows from the previous theorem. The previous theorem deals with symmetries of a selfadjoint operator L. Often,theselfadjoint operator arises as the Friedrichs extension of a symmetric operator. We next study how symmetries of a symmetric operator carry over to its Friedrichs extension. Specifically, we consider the following situation: (*) Let H be a Hilbert space with inner product,. LetL 0 be a symmetric operator on H with domain D 0.LetQ 0 be the associated form, i.e., Q 0 is defined on D 0 D 0 via Q 0 (u,v) := L 0 u,v. Assume that Q 0 is non-negative, i.e., Q 0 (u, u) 0forall u D 0. Then, Q 0 is closable. Let Q be the closure of Q 0, D(Q) the domain of Q and L the Friedrichs extension of L 0, i.e., L is the selfadjoint operator associated to Q. Lemma A.5 Assume ( ). Let A be a bounded operator on H with D 0 invariant under A and A,AL 0 x = L 0 Ax and A L 0 x = L 0 A x for all x D 0. Then, the following assertions are equivalent: (i) D(L) is invariant under A and AL x = L Ax for all x D(L). (ii) D(Q) is invariant under A and A and Q(Ax, y) = Q(x, A y) for all x, y D(Q). (iii) There exists a C 0 with both Q 0 (Ax, Ax) CQ 0 (x, x) and Q 0 (A x, A x) CQ 0 (x, x) for all x D 0. Proof (iii) (ii): By AL 0 x = L 0 Ax for all x D 0 we infer that Q 0 (Ax, y) = Q 0 (x, A y) for all x, y D 0.AsQ is the closure of Q 0, it now suffices to show that both (Au n ) and (A u n ) are a Cauchy sequences with respect to the Q-norm, whenever (u n ) is a Cauchy sequence with respect to the Q-norm in D 0. This follows directly from (iii). (ii) (i): Let x D(L) be given. Then, x belongs to D(Q) and, by (ii), Ax belongs to D(Q) as well. Thus, we can calculate for all y D(Q) Q(Ax, y) = Q(x, A y) = Lx, A y = ALx, y. This implies that Ax D(L) and LAx = ALx. Hence, we obtain (i). (i) (iii): From Theorem A.1 and (i) we infer that L 1/2 Ax = AL 1/2 x for all x D(L 1/2 ).Now,forx D 0 it holds that Q 0 (x, x) = L 0 x, x = L 1/2 x, L 1/2 x = L 1/2 x 2. By Ax D 0 for x D 0 a direct calculation gives Q 0 (Ax, Ax) = L 1/2 Ax 2 = AL 1/2 x 2 A 2 L 1/2 x 2 = A 2 Q 0 (x, x). A similar argument shows Q 0 (A x, A x) A 2 Q 0 (x, x). This finishes the proof. 123

166 Volume growth, spectrum and stochastic completeness 929 We now turn to the special situation that A is the projection onto a closed subspace. In this case, some further strengthening of the above result is possible. We first provide an appropriate definition. Definition A.6 Let H be a Hilbert space and S a symmetric operator on H with domain D(S). A closed subspaceu of H with associated orthogonal projection P is called a reducing subspace for S if D(S) is invariant under P and SPx = PSPx for all x D(S). The previous definition is just a commutation condition in the form discussed above as shown in the next lemma. Lemma A.7 Let S be a symmetric operator on the Hilbert space H and P be the orthogonal projection onto a closed subspace U of H. Then, the following assertions are equivalent: (i) U is a reducing subspace for S. (ii) D(S) is invariant under P and S Px = P Sx holds for all x D(S). Proof The implication (ii) (i) is obvious. It remains to show (i) (ii): We first show PSy = 0forally D(S) with Py = 0 (i.e., y U): Choose x D(S) arbitrarily. Then, as Px D(S) D(S ) we obtain PSy, x = Sy, Px = y, S Px = y, SPx = y, PSPx = Py, SPx =0. As D(S) is dense, we infer PSy = 0. Let now x D(S) be arbitrary. Then, x = Px + (1 P)x and both Px and (1 P)x belong to D(S). Thus, we can calculate PSx = PSPx + PS(1 P)x = PSPx = SPx. This finishes the proof. We now come to the main result of the appendix dealing with symmetries of symmetric operators in terms of reducing subspaces. Corollary A.8 Assume ( ). LetU beaclosedsubspaceofh and A theorthogonalprojection onto U. Assume that D 0 is invariant under A. Then, the following assertions are equivalent: (i) U is a reducing subspace for L 0, i.e., L 0 Ax = AL 0 x for all x D 0. (ii) Q 0 (Ax, Ay) = Q 0 (Ax, y) = Q 0 (x, Ay) for all x, y D 0. (iii) D(Q) is invariant under A and Q(Ax, Ay) = Q(Ax, y) = Q(x, Ay) for all x, y D(Q). (iv) U is a reducing subspace for L. (v) A commutes with e tl for every t 0. (vi) A commutes with (L + α) 1 for any α>0. Proof Obviously, (i) and (ii) are equivalent. The equivalence of (iii) and (iv) follows from the equivalence of (i) and (ii) in Lemma A.5. The equivalence between (iv), (v) and (vi) follows immediately from Theorem A.1. The implication (iii) (ii) is clear (as AD 0 D 0 ). It remains to show (ii) (iii): A direct calculation using (ii) gives for all x D 0 that Q 0 (x, x) = Q 0 ((A + (1 A))x, x) = Q 0 (Ax, x) + Q 0 ((1 A)x, x) = Q 0 (Ax, Ax) + Q 0 ((1 A)x,(1 A)x). 123

167 930 M. Keller et al. This shows Q 0 (Ax, Ax) Q 0 (x, x) for all x D 0. Now, the implication (iii) (ii) from Lemma A.5 gives (iii). References 1. Bär, C., Pacelli Bessa, G.: Stochastic completeness and volume growth. Proc. Am. Math. Soc. 138(7), (2010). doi: /s Barroso, C.S., Pacelli Bessa, G.: Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds. Int. J. Appl. Math. Stat. 6(D06), (2006) 3. Bauer, F., Jost, J., Liu, S.: Ollivier Ricci curvature and the spectrum of the normalized graph Laplace operator (2012, preprint). arxiv: v1[math.co] 4. Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discret. Comput. Geom. 25(1), (2001) 5. Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), (2006). doi: /advgeom Beurling, A., Denny, J.: Espaces de Dirichlet. I. Le cas élémentaire. Acta Math. 99, (1958) (French) 7. Biggs, N.L., Mohar, B., Shawe-Taylor, J.: The spectral radius of infinite graphs. Bull. London Math. Soc. 20(2), (1988). doi: /blms/ Bonciocat, A.-I., Sturm, K.-T.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), (2009). doi: /j.jfa Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178(4), (1981). doi: /bf Brooks, R.: The spectral geometry of k-regular graphs. J. Anal. Math. 57, (1991) 11. Chavel, I., Karp, L.: Large time behavior of the heat kernel: the parabolic λ-potential alternative. Comment. Math. Helv. 66(4), (1991). doi: /bf Cheeger, J., Yau, S.T.: A lower bound for the heat kernel. Commun. Pure Appl. Math. 34(4), (1981). doi: /cpa Cheng, S.Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(3), (1975) 14. Dodziuk, J.: Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ.Math.J.32(5), (1983). doi: /iumj Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am.Math.Soc.284(2), (1984). doi: / Dodziuk, J.: Elliptic operators on infinite graphs. Analysis, geometry and topology of elliptic operators. World Sci. Publ., Hackensack NJ pp (2006) 17. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math., vol. 73. Amer. Math. Soc., Providence, pp (1988) 18. Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, pp (1986) 19. Dodziuk, J., Mathai, V.: Kato s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel, Contemp. Math., vol. 398, Amer. Math. Soc., Providence RI, pp (2006) 20. Feller, W.: On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 65(2), (1957) 21. Forman, R.: Bochner s method for cell complexes and combinatorial Ricci curvature. Discret. Comput. Geom. 29(3), (2003). doi: /s x 22. Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. (2) 48(2), , (1996). doi: /tmj/ Fujiwara, K.: The Laplacian on rapidly branching trees. Duke Math. J. 83(1), (1996). doi: /S Fukushima, M., Ōshima, Y., Masayoshi Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin (1994) 123

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171 CHAPTER 7 M. Bonnefont, S. Golénia, M. Keller, Eigenvalue asymptotics for Schrödinger operators on sparse graphs, arxiv:

172 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS MICHEL BONNEFONT, SYLVAIN GOLÉNIA, AND MATTHIAS KELLER Abstract. We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics. 1. Introduction The spectral theory of discrete Laplacians on finite or infinite graphs has drawn a lot of attention for decades. One important aspect is to understand the relations between the geometry of the graph and the spectrum of the Laplacian. Often a particular focus lies on the study of the bottom of the spectrum and the eigenvalues below the essential spectrum. Certainly the most well-known estimates for the bottom of the spectrum of Laplacians on infinite graphs are so called isoperimetric estimates or Cheeger inequalities. Starting with [D1] in the case of infinite graphs, these inequalities were intensively studied and resulted in huge body of literature, where we here mention only [BHJ, BKW, D2, DK, F, M1, M2, K1, KL2, Woj1]. In certain more specific geometric situations the bottom of the spectrum might be estimated in terms of curvature, see [BJL, H, JL, K1, K2, KP, LY, Woe]. There are various other more recent approaches such as Hardy inequalities in [G] and summability criteria involving the boundary and volume of balls in [KLW]. In this work we focus on sparse graphs to study discreteness of spectrum and eigenvalue asymptotics. In a moral sense, the term sparse means that there are not too many edges, however, throughout the years various different definitions were investigated. We mention here [EGS, L] as seminal works which are closely related to our definitions. As it is impossible to give a complete discussion of the development, we refer to some selected more recent works such as [AABL, B, LS, M2] and references therein which also illustrates the great variety of possible definitions. Here, we discuss three notions of sparseness that result in a hierarchy of very general classes of graphs. Let us highlight the work of Mohar [M3], where large eigenvalues of the adjacency matrix on finite graphs are studied. Although our situation of infinite graphs Date: January 24, Mathematics Subject Classification. 47A10, 34L20,05C63, 47B25, 47A63. Key words and phrases. discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, planarity, sparse, functional inequality. 1

173 2 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER with unbounded geometry requires fundamentally different techniques functional analytic rather than combinatorial in spirit our work is certainly closely related. The techniques used in this paper owe on the one hand to considerations of isoperimetric estimates as well as a scheme developed in [G] for the special case of trees. In particular, we show that a notion of sparseness is a geometric characterization for an inequality of the type (1 a) deg k (1 + a) deg +k for some a (0, 1), k 0 which holds in the form sense (precise definitions and details will be given below). The moral of this inequality is that the asymptotic behavior of the Laplacian is controlled by the vertex degree function deg (the smaller a the better the control). Furthermore, such an inequality has very strong consequences which follow from well-known functional analytic principles. These consequences include an explicit description of the form domain, characterization for discreteness of spectrum and eigenvalue asymptotics. Let us set up the framework. Here, a graph G is a pair (V, E ), where V denotes a countable set of vertices and E : V V {0, 1} is a symmetric function with zero diagonal determining the edges. We say two vertices x and y are adjacent or neighbors whenever E (x, y) = E (y, x) = 1. In this case, we write x y and we call (x, y) and (x, y) the (directed) edges connecting x and y. We assume that G is locally finite that is each vertex has only finitely many neighbors. For any finite set W V, the induced subgraph G W := (W, E W ) is defined by setting E W := E W W, i.e., an edge is contained in G W if and only if both of its vertices are in W. We consider the complex Hilbert space l 2 (V ) := {ϕ : V C such that x V ϕ(x) 2 < } endowed with the scalar product ϕ, ψ := x V ϕ(x)ψ(x), ϕ, ψ l 2 (V ). For a function g : V C, we denote the operator of multiplication by g on l 2 (V ) given by ϕ gϕ and domain D(g) := {ϕ l 2 (V ) gϕ l 2 (V )} with slight abuse of notation also by g. Let q : V [0, ). We consider the Schrödinger operator + q defined as { ( D( + q) := ϕ l 2 (V ) v w v(ϕ(v) ) } ϕ(w)) + q(v)ϕ(v) l 2 (V ) ( + q)ϕ(v) := w v(ϕ(v) ϕ(w)) + q(v)ϕ(v). The operator is non-negative and selfadjoint as it is essentially selfadjoint on C c (V ), the set of finitely supported functions V R, (confer [Woj1, Theorem 1.3.1], [KL1, Theorem 6]). In Section 2 we will allow for potentials whose negative part is form bounded with bound strictly less than one. Moreover, in Section 4 we consider also magnetic Schrödinger operators. As mentioned above sparse graphs have already been introduced in various contexts with varying definitions. In this article we also treat various natural generalizations of the concept. In this introduction we stick to an intermediate situation. Definition. A graph G := (V, E ) is called k-sparse if for any finite set W V the induced subgraph G W := (W, E W ) satisfies 2 E W k W,

174 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 3 where A denotes the cardinality of a finite set A and we set E W := 1 2 {(x, y) W W E W (x, y) = 1}, that is we count the non-oriented edges in G W. Examples of sparse graphs are planar graphs and, in particular, trees. We refer to Section 6 for more examples. For a function g : V R and a finite set W V, we denote Moreover, we define lim inf g(x) := sup x g(w ) := x W inf g(x), W V finite x V \W g(x). lim sup g(x) := x inf sup W V finite x V \W g(x). For two selfadjoint operators T 1, T 2 on a Hilbert space and a subspace D 0 D(T 1 ) D(T 2 ) we write T 1 T 2 on D 0 if T 1 ϕ, ϕ T 2 ϕ, ϕ for all ϕ D 0. Moreover, for a selfadjoint semi-bounded operator T on a Hilbert space, we denote the eigenvalues below the essential spectrum by λ n (T ), n 0, with increasing order counted with multiplicity. The next theorem is a special case of the more general Theorem 2.2 in Section 2. It illustrates our results in the case of sparse graphs introduced above and includes the case of trees, [G, Theorem 1.1], as a special case. While the proof in [G] uses a Hardy inequality, we rely on some new ideas which have their roots in isoperimetric techniques. The proof is given in Section 2.2. Theorem 1.1. Let G := (V, E ) be a k-sparse graph and q : V [0, ). Then, we have the following: (a) For all 0 < ε 1, (1 ε)(deg +q) k 2 ( ) 1 ε ε + q (1 + ε)(deg +q) + k ( ) 1 2 ε ε, on C c (V ). (b) D ( ( + q) 1/2) = D ( (deg + q) 1/2). (c) The operator + q has purely discrete spectrum if and only if In this case, we obtain lim inf(deg + q)(x) =. x lim inf λ λ n ( + q) λ n (deg + q) = 1. As a corollary, we obtain following estimate for the bottom and the top of the (essential) spectrum. Corollary 1.2. Let G := (V, E ) be a k-sparse graph and q : V [0, ). Define d := inf x V (deg +q)(x) and D := sup x V (deg +q)(x). Assume d < k D < +, then d 2 k 2 ( d k ) ( k inf σ( + q) sup σ( + q) D 2 D k )

175 4 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER Define d ess := lim inf x (deg +q)(x) and D ess := lim sup x (deg +q)(x). Assume d ess < k D ess < +, then ( k d ess 2 d ess k ) ( k inf σ ess ( +q) sup σ ess ( +q) D ess 2 D ess k ) ( ) k Proof of Corollary 1.2. The conclusion follows by taking ε = min 2d k, 1 in (a) in of Theorem 1.1. Remark 1.3. The bounds in Corollary 1.2 are optimal for the bottom and the top of the (essential) spectrum in the case of regular trees. The paper is structured as follows. In the next section an extension of the notion of sparseness is introduced which is shown to be equivalent to a functional inequality and equality of the form domains of and deg. In Section 3 we consider almost sparse graphs for which we obtain precise eigenvalue asymptotics. Furthermore, in Section 4 we shortly discuss magnetic Schrödinger operators. Our notion of sparseness has very explicit but non-trivial connections to isoperimetric inequalities which are made precise in Section 5. Finally, in Section 6 we discuss some examples. 2. A geometric characterization of the form domain In this section we characterize equality of the form domains of + q and deg +q by a geometric property. This geometric property is a generalization of the notion of sparseness from the introduction. Before we come to this definition, we introduce the class of potentials that is treated in this paper. Let α > 0. We say a potential q : V R is in the class K α if there is C α 0 such that q α( + q + ) + C α, where q ± := max(±q, 0). For α (0, 1), we define the operator + q via the form sum of the operators + q + and q (i.e., by the KLMN Theorem, see e.g., [RS, Theorem X.17]). Note that + q is bounded from below and D( + q 1 2 ) = D(( + q+ ) ) = D( 2 ) D(q 2 +), where + q is defined by the spectral theorem. The last equality follows from [GKS, Theorem 5.6]. in the sense of functions and forms. An other important class are the potentials K 0 + := K α. α (0,1) In our context of sparseness, we can characterize the class K 0 + to be the potentials whose negative part q is morally o(deg +q + ), see Corollary 2.9. Let us mention that if q is in the Kato class with respect to + q +, i.e., if we have lim sup t 0 + e t( +q+) q = 0, then q := q + q K 0 + by [SV, Theorem 3.1]. Next, we come to an extension of the notion of sparseness. For a set W V, let the boundary W of W be the set of edges emanating from W W := {(x, y) W V \ W x y}.

176 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 5 Definition. Let G := (V, E ) be a graph and q : V R. For given a 0 and k 0, we say that (G, q) is (a, k)-sparse if for any finite set W V the induced subgraph G W := (W, E W ) satisfies 2 E W k W + a( W + q + (W )). Remark 2.1. (a) Observe that the definition depends only on q +. The negative part of q will be taken in account through the hypothesis K α or K 0 + in our theorems. (b) If (G, q) is (a, k)-sparse, then (G, q ) is (a, k)-sparse for every q q. (c) As mentioned above there is a great variety of definitions which were so far predominantly established for (families of) finite graphs. For example it is asked that E = C V in [EGS], E W k W + l in [L, LS], E O( V ) in [AABL] and deg(w ) k W in [M3]. We now characterize the equality of the form domains in geometric terms. Theorem 2.2. Let G := (V, E ) be a graph and q K α, α (0, 1). The following assertions are equivalent: (i) There are a, k 0 such that (G, q) is (a, k)-sparse. (ii) There are ã (0, 1) and k 0 such that on C c (V ) (1 ã)(deg +q) k + q (1 + ã)(deg +q) + k. (iii) There are ã (0, 1) and k 0 such that on C c (V ) (iv) D( + q 1/2 ) = D( deg +q 1/2 ). (1 ã)(deg +q) k + q. Furthermore, + q has purely discrete spectrum if and only if In this case, we obtain 1 ã lim inf n lim inf(deg + q)(x) =. x λ n ( + q) lim sup λ n (deg +q) n λ n ( + q) λ n (deg +q) 1 + ã. Before we come to the proof of Theorem 2.2, we summarize the relation between the sparseness parameters (a, k) and the constants (ã, k) in the inequality in Theorem 2.2 (ii). Remark 2.3. Roughly speaking a tends to as ã tends to 1 and a tends to 0 + as ã tends to 0 + and vice-versa. More precisely, Lemma 2.5 we obtain that for given ã and k the values of a and k can be chosen to be a = ã 1 ã and k = k 1 ã. Reciprocally, given a, k 0 and q : V [0, ), Lemma 2.7 distinguishes the case where the graph is sparse a = 0 and a > 0. For a = 0 we may choose ã (0, 1) arbitrary and k = k ( 1 ) 2 ã ã.

177 6 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER For an (a, k)-sparse graph with a > 0 the precise constants are found below in Lemma 2.7. Here, we discuss the asymptotics. For a 0 +, we obtain ã k 2a and k 2a, and for a ã 1 3 3k 8a 2 and k 4a. In the case q K α, the constants ã, k from the case q 0 have to be replaced by constants whose formula can be explicitly read from Lemma A.3. For α 0 +, the constant replacing ã tends to ã while the asymptotics of the constant replacing k depend also on the behavior of C α from the assumption q α( + q) + C α. Remark 2.4. (a) Observe that in the context of Theorem 2.2 statement (iv) is equivalent to (iv ) D( + q 1/2 ) = D((deg +q + ) 1/2 ). Indeed, (ii) implies the corresponding inequality for q = q +. Thus, as q K α, D( + q 1 2 ) = D(( + q+ ) 1 2 ) = D((deg +q+ ) 1 2 ). (b) The definition of the class K 0 + is rather abstract. Indeed, Theorem 2.2 yields a very concrete characterization of these potentials, see Corollary 2.9 below. (c) Theorem 2.2 characterizes equality of the form domains. Another natural question is under which circumstances the operator domains agree. For a discussion on this matter we refer to [G, Section 4.1]. The rest of this section is devoted to the proof of the results which are divided into three parts. The following three lemmas essentially show the equivalences (i) (ii) (iii) providing the explicit dependence of (a, k) on (ã, k) and vice versa. The third part uses general functional analytic principles collected in the appendix. The first lemma shows (iii) (i). Lemma 2.5. Let G := (V, E ) be a graph and q : V R. If there are ã (0, 1) and k 0 such that for all f in C c (V ), then (G, q) is (a, k)-sparse with (1 ã) f, (deg +q)f k f 2 f, f + qf, a = ã 1 ã and k = k 1 ã. Remark 2.6. We stress that we suppose solely that q : V R and work with Cc(V ) + q Cc(V ). We do not specify any self-adjoint extension of the latter. Proof. Let f C c (V ). By adding q to the assumed inequality we obtain immediately (1 ã) f, (deg +q + )f k f 2 f, f + q + f. Let W V be a finite set and denote by 1 W the characteristic function of the set W. We recall the basic equalities deg(w ) = 2 E W + W and 1 W, 1 W = W.

178 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 7 Therefore, applying the asserted inequality with f = 1 W, we obtain 2 E W k 1 ã W + ã 1 ã ( W + q +(W )). This proves the statement. The second lemma gives (i) (ii) for q 0. Lemma 2.7. Let G := (V, E ) be a graph and q : V [0, ). If there are a, k 0 such that (G, q) is (a, k)-sparse, then (1 ã)(deg +q) k + q (1 + ã)(deg +q) + k. on C c (V ), where if (G, q) is sparse, i.e., a = 0, we may choose ã (0, 1) arbitrary and k = k ( 1 ) 2 ã ã. In the other case, i.e. a > 0, we may choose min ( 1 4 ã =, a2) ( + 2a + a 2 ( max 3 and k = max 2, 1 a a) ) k, 2k(1 ã). (1 + a) 2(1 + a) Proof. Let f C c (V ) be complex valued. Assume first that f, (deg +q)f < k f 2. In this case, remembering 2 deg, we can choose ã (0, 1) arbitrary and k such that k 2(1 ã)k. So, assume f, (deg +q)f k f 2. Using an area and a co-area formula (cf. [KL2, Theorem 12 and Theorem 13]) with Ω t := {x V f(x) 2 > t}, in the first step and the assumption of sparseness in the third step, we obtain ( ) f,(deg +q)f k f 2 = deg(ω t ) + q(ω t ) k Ω t dt = 0 (1 + a) = (1 + a) 2 0 ( ) 2 E Ωt + Ω t + q(ω t ) k Ω t dt 0 x,y,x y Ω t + q(ω t )dt f(x) 2 f(y) 2 + (1 + a) q(x) f(x) 2 (1 + a) (f(x) f(y))(f(x) + f(y)) + (1 + a) q(x) f(x) 2 2 x,y,x y x ( (1 + a) f(x) f(y) ) 1/2 q(x) f(x) 2 2 x,y,x y x ( f(x) + f(y) ) 1/2 q(x) f(x) 2 x,y,x y x ( ) = (1 + a) f, ( + q)f f, (deg +q)f f, ( + q)f 2, x

179 8 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER where we used the Cauchy-Schwarz inequality in the last inequality and basic algebraic manipulation in the last equality. Since the left hand side is non-negative by the assumption f, (deg +q)f k f 2, we can take square roots on both sides. To shorten notation, we assume for the rest of the proof q 0 since the proof with q 0 is completely analogous. Reordering the terms, yields (1 + a) 2 f, f 2 2(1 + a) 2 f, deg f f, f + ( f, (deg k)f ) 2 0. Resolving the quadratic expression above gives, with f, deg f δ f, f f, deg f + δ, δ := f, deg f 2 (1 + a) 2 ( f, (deg k)f ) 2. Using 4ξζ (ξ + ζ) 2, ξ, ζ 0, for all 0 < λ < 1, we estimate δ as follows (1 + a) 2 δ = (2a + a 2 ) f, deg f 2 + k f 2 f, (2 deg k)f ( (2a + a 2 ) f, deg f 2 + λ f, deg f + k ( ) ) λ λ f 2 ( λ2 + 2a + a 2 f, deg f + k ( ) ) λ λ f 2. If a = 0, i.e., the k-sparse case, then we take λ = ã to get ( δ k f 2 f, 2 deg f ã f, deg f + k ( ) 1 2 ã ã f 2) 2. As k/2ã 2(1 ã)k, this proves the desired inequality with k = k/2ã. If a > 0, we take λ = min ( 1 2, a) to get (( ( ) ) 1 (1 + a) 2 δ = min 4, a2 + 2a + a 2 f, deg f + k2 ( ( )) )2 32 1a max, a f 2. Keeping in mind the restriction k 2(1 ã)k for the case f, (deg +q)f < k f 2, this gives the statement with the choice of (ã, k) in the statement of the lemma. The two lemmas above are sufficient to prove Theorem 2.2 for the case q 0. An application of Lemma A.3 turns the lower bound of Lemma 2.7 into a corresponding lower bound. This straightforward argument does not work for the upper bound. However, the following surprising lemma shows that such a lower bound by deg automatically implies the corresponding upper bound. There is a deeper reason for this fact which shows up in the context of magnetic Schrödinger operators. We present the non-magnetic version of the statement here for the sake of being self-contained in this section. For the more conceptual and more general magnetic version, we refer to Lemma 4.4. Lemma 2.8 (Upside-Down-Lemma non-magnetic version). Let G := (V, E ) be a graph and q : V R. Assume there are ã (0, 1), k 0 such that for all f C c (V ), (1 ã) f, (deg +q)f k f 2 f, f + qf,

180 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 9 then for all f C c (V ), we also have f, f + qf (1 + ã) f, (deg +q)f + k f 2. Proof. By a direct calculation we find for f C c (V ) f, (2 deg )f = 1 (2 f(x) f(y) 2 ) f(x) f(y) 2 ) 2 = 1 2 x,y V,x y x,y,x y = f, f. f(x) + f(y) x,y,x y f(x) f(y) 2 Adding q to the inequality and using the assumption gives after reordering f, ( + q)f 2 f, (deg +q)f f, ( + q) f which yields the assertion. (1 ã) f, (deg +q) f + k f, f = (1 ã) f, (deg +q)f + k f, f Proof of Theorem 2.2. The implication (i) (iii) follows from Lemma 2.7 applied with q + and from Lemma A.3 with q. The implication (iii) (ii) follows from the Upside-Down-Lemma above. Furthermore, (ii) (i) is implied by Lemma 2.5. The equivalence (ii) (iv) follows from an application of the Closed Graph Theorem, Theorem A.1. Finally, the statements about discreteness of spectrum and eigenvalue asymptotics follow from an application of the Min-Max-Principle, Theorem A.2. Proof of Theorem 1.1. (a) follows from Lemma 2.7. The other statements follow directly from Theorem 2.2. As a corollary we can now determine the potentials in the class K 0 + explicitly and give necessary and sufficient criteria for potentials being in K α, α (0, 1). Corollary 2.9. Let (G, q) be an (a, k)-sparse graph for some a, k 0. (a) The potential q is in K 0 + if and only if for all α (0, 1) there is κ α 0 such that q α(deg +q + ) + κ α. (b) Let α (0, 1) and ã = min(1/4, a 2 ) + 2a + a 2 /(1 + a) (as given by Lemma 2.7). If there is κ α 0 such that q α(deg +q + ) + κ α, then q K α/(1 ã). On the other, hand if q K α, then there is κ α 0 such that q α(1 + ã)(deg +q + ) + κ α. Proof. Using the assumption q α(deg +q ) + κ α and the lower bound of Theorem 2.2 (ii), we infer α q α(deg +q + ) + κ α (1 ã) ( + q +) + α (1 ã) k + κ α. Conversely, q K α and the upper bound of Theorem 2.2 (ii) yields q α( + q + ) + C α α(1 + ã)(deg +q + ) + α k + C α. Hence, (a) follows. For (b), notice that ã = min(1/4, a 2 ) + 2a + a 2 /(1 + a) by Lemma 2.7.

181 10 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER 3. Almost-sparseness and asymptotic of eigenvalues In this section we prove better estimates on the eigenvalue asymptotics in a more specific situation. Looking at the inequality in Theorem 2.2 (ii) it seems desirable to have ã = 0. As this is impossible when the degree is unbounded, we consider a sequence of ã that tends to 0. Keeping in mind Remark 2.3, this leads naturally to the following definition. Definition. Let G := (V, E ) be a graph and q : V R. We say (G, q) is almost sparse if for all ε > 0 there is k ε 0 such that (G, q) is (ε, k ε )-sparse, i.e., for any finite set W V the induced subgraph G W := (W, E W ) satisfies 2 E W k ε W + ε ( W + q + (W )). Remark 3.1. (a) Every sparse graph G is almost sparse. (b) For an almost sparse graph (G, q), every graph (G, q ) with q q is almost sparse. The main result of this section shows how the first order of the eigenvalue asymptotics in the case of discrete spectrum can be determined for almost sparse graphs. Theorem 3.2. Let G := (V, E ) be a graph and q K 0 +. The following assertions are equivalent: (i) (G, q) is almost sparse. (ii) For every ε > 0 there are k ε 0 such that on C c (V ) (1 ε)(deg +q) k ε + q (1 + ε)(deg +q) + k ε. (iii) For every ε > 0 there are k ε 0 such that on C c (V ) (1 ε)(deg +q) k ε + q. Moreover, D(( +q) 1/2 ) = D((deg +q) 1/2 ) and the operator +q has purely discrete spectrum if and only if lim inf x (deg + q)(x) =. In this case, we have lim n λ n ( + q) λ n (deg +q) = 1. Proof. The statement is a direct application of Theorem 2.2 if one keeps track of the constants given explicitly by Lemma 2.5, Lemma 2.7 and Lemma A Magnetic Laplacians In this section, we consider magnetic Schrödinger operators. Clearly, every lower bound can be deduced from Kato s inequality. However, for the eigenvalue asymptotics we also need to prove an upper bound. We fix a phase θ : V V R/2πZ such that θ(x, y) = θ(y, x). For a potential q : V [0, ) we consider the magnetic Schrödinger operator θ + q defined as { ( D( θ + q) := ϕ l 2 (V ) v x y(ϕ(x) ) } e iθ(x,y) ϕ(x)) + q(x)ϕ(x) l 2 (V ) ( θ + q)ϕ(x) := x y(ϕ(x) e iθ(x,y) ϕ(y)) + q(x)ϕ(x).

182 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 11 A computation for ϕ C c (V ) gives ϕ, ( θ + q)ϕ = 1 ϕ(x) e iθ(x,y) ϕ(y) 2 + q(x) ϕ(x) 2. 2 x x,y,x y The operator is non-negative and selfadjoint as it is essentially selfadjoint on C c (V ) (confer e.g. [G]). For α > 0, let Kα θ be the class of real-valued potentials q such that q α( θ + q + ) + C α for some C α 0. Denote K0 θ = K θ + α. α (0,1) Again, for α (0, 1) and q K θ α, we define θ + q to be the form sum of θ + q + and q. We present our result for magnetic Schrödinger operators which has one implication from the equivalences of Theorem 2.2 and Theorem 3.2. Theorem 4.1. Let G := (V, E ) be a graph, θ be a phase and q K θ 0 + be a potential. Assume (G, q) is (a, k)-sparse for some a, k 0. Then, we have the following: (a) There are ã (0, 1), k 0 such that on C c (V ) (1 ã)(deg +q) k θ + q (1 + ã)(deg +q) + k. (b) D ( θ + q 1/2) = D ( deg + q 1/2). (c) The operator θ + q has purely discrete spectrum if and only if lim inf(deg + q)(x) =. x In this case, if (G, q) is additionally almost sparse, then lim inf λ λ n ( θ + q) λ n (deg + q) = 1. Remark 4.2. (a) The constants ã and k can chosen to be the same as the ones we obtained in the proof of Theorem 2.2, i.e., these constants are explicitly given combining Lemma 2.7 and Lemma A.3. (b) Statement (a) and (b) of the theorem above remain true for q K α, α (0, 1) since K α Kα θ by Kato s inequality below. We will prove the theorem by applying Theorem 2.2 and Theorem 3.2. The considerations heavily rely on Kato s inequality and a conceptual version of the Upside-Down-Lemma, Lemma 2.8, which shows that a lower bound for +q implies an upper and lower bound on θ + q. Secondly, in Theorem 3.2 potentials in K 0 + are considered, while here we start with the class K0 θ. However, it can be seen + that K 0 + = K0 θ in the case of (a, k)-sparse graph, see Lemma 4.5 below. + As mentioned above a key fact is Kato s inequality, see e.g. [DM, Lemma 2.1] or [GKS, Theorem 5.2.b]. Proposition 4.3 (Kato s inequality). Let G := (V, E ) be a graph, θ be a phase and q : V R. For all f C c (V), we have In particular, for all α > 0 f, ( f + q f ) f, ( θ f + qf). K α K θ α and K 0 + K θ 0 +.

183 12 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER Proof. The proof of the inequality can be obtained by a direct calculation. The second statement is an immediate consequence. The next lemma is a rather surprising observation. It is the magnetic version of the Upside-Down-Lemma, Lemma 2.8. Lemma 4.4 (Upside-Down-Lemma magnetic version). Let G := (V, E ) be a graph, θ be a phase and q : V R be a potential. Assume that there are ã (0, 1) and k 0 such that for all f C c (V ), we have then for all f C c (V ), we also have (1 ã) f, (deg +q)f k f 2 f, f + qf (1 ã) f, (deg +q)f k f 2 f, θ f + qf (1 + ã) f, (deg +q)f + k f 2. Proof. The lower bound follows directly from Kato s inequality and the lower bound from the assumption (since f, (deg +q)f = f, (deg +q) f for all f C c (V )). Now, observe that for all θ θ = 2 deg θ+π. So, the upper bound for θ + q follows from the lower bound of θ+π + q which we deduced from Kato s inequality. The lemma above allows to relate the classes K α and K θ α for (a, k)-sparse graphs. Lemma 4.5. For a, k 0 let G := (V, E ) be an (a, k)-sparse graph, θ be a phase and α > 0. Then, K θ α K α, for α = 1 + ã 1 ã α, where ã is given in Lemma 2.7. In particular, Moreover, if (G, q) is almost-sparse, then K θ 0 + = K 0 +. K θ α K α, for all α > α. Proof. Let q K θ α. Applying Lemma 2.7, we get + q + (1 ã)(deg +q + ) k. Now, by the virtue of the Upside-Down-Lemma, Lemma 4.4, we infer θ + q + (1 + ã)(deg +q + ) + k 1 + ã 1 ã ( + q +) ã k which implies the first statement and K0 θ K The reverse inclusion K0 θ + K 0 + follows from Kato s inequality, Lemma 4.3. For almost sparse graphs a can be chosen arbitrary small and accordingly ã (from Lemma 2.7) becomes arbitrary small. Hence, the statement Kα θ K α, for α > α follows from the inequality above. Proof of Theorem 4.1. Let q K0 θ. By Lemma 4.5, q K Thus, (a) follows from Theorem 2.2 and Lemma 4.4. Using (a) statement (b) follows from an application of the Closed Graph Theorem, Theorem A.1 and statement (c) follows from an application of the Min Max Principle, Theorem A.2.

184 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 13 Remark 4.6. Instead of using Kato s inequality one can also reproduce the proof of Lemma 2.7 using the following estimate f(x) 2 f(y) 2 (f(x) e iθ(x,y) f(y))(f(x) + e iθ(x,y) f(y)). So, we infer the key estimate: f,(deg +q)f k f 2 (1 + a) f, ( θ + q)f 1 2 f, ( θ+π + q)f 1 2, = (1 + a) f, ( θ + q)f 1 2 ( 2 f, (deg +q)f f, ( θ + q)f ) 1 2. The rest of the proof is analogous. It can be observed that unlike in Theorem 2.2 or Theorem 3.2 we do not have an equivalence in the theorem above. A reason for this seems to be that our definition of sparseness does not involve the magnetic potential. This direction shall be pursued in the future. Here, we restrict ourselves to some remarks on the perturbation theory in the context of Theorem 4.1 above. Remark 4.7. (a) If the inequality Theorem 4.1 (a) holds for some θ, then the inequality holds with the same constants for θ and θ ± π. This can be seen by the fact θ+π = 2 deg and f, θ f = f, θ f while f, deg f = f, deg f for f C c (V ). (b) The set of θ such that Theorem 4.1 (a) holds true for some fixed ã and k is closed in the product topology, i.e., with respect to pointwise convergence. This follows as f, θn f f, θ f if θ n θ, n, for fixed f C c (V ). (c) For two phases θ and θ let h(x) = max y x θ(x, y) θ (x, y). By a straight forward estimate lim sup x h(x) = 0 implies that for every ε > 0 there is C 0 such that ε deg C θ θ ε deg +C on C c (V ). We discuss three consequences of this inequality: First of all, this inequality immediately yields that if D( 1/2 θ ) = D(deg 1/2 ) then D( 1/2 θ ) = D(deg1/2 ) (by the KLMN Theorem, see e.g., [RS, Theorem X.17]) which in turn yields equality of the form domains of θ and θ. Secondly, combining this inequality with Theorem 3.2 we obtain the following: If lim sup x max y x θ(x, y) = 0 and for every ε > 0 there is k ε 0 such that (1 ε) deg k ε θ (1 + ε) deg +k ἐ then the graph is almost sparse and in consequence the inequality in Theorem 4.1 (a) holds for any phase. Thirdly, using the techniques in the proof of [G, Proposition 5.2] one shows that the essential spectra of θ and θ coincide. With slightly more effort and the help of the Kuroda-Birman Theorem, [RS, Theorem XI.9] one can show that if h l 1 (V ), then even the absolutely continuous spectra of θ and θ coincide. 5. Isoperimetric estimates and sparseness In this section we relate the concept of sparseness with the concept of isoperimetric estimates. First, we present a result which should be viewed in the light of Theorem 2.2 as it points out in which sense isoperimetric estimates are stronger

185 14 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER than our notions of sparseness. In the second subsection, we present a result related to Theorem 3.2. Finally, we present a concrete comparison of sparseness and isoperimetric estimates. As this section is of a more geometric flavor we restrict ourselves to the case of potentials q : V [0, ) Isoperimetric estimates. Let U V and define the Cheeger or isoperimetric constant of U by α U := inf W U finite W + q(w ) deg(w ) + q(w ). In the case where deg(w ) + q(w ) = 0, for instance when W is an isolated point, by convention the above quotient is set to be equal to 0, Note that α U [0, 1). The following theorem illustrates in which sense positivity of the Cheeger constant is linked with (a, 0)-sparseness. We refer to Theorem 5.4 for precise constants. Theorem 5.1. Given G := (V, E ) a graph and q : V [0, ). The following assertions are equivalent (i) α V > 0. (ii) There is ã (0, 1) (1 ã)(deg +q) + q (1 + ã)(deg +q). (iii) There is ã (0, 1) such that (1 ã)(deg +q) + q. The implication (iii) (i) is already found in [G, Proposition 3.4]. The implication (i) (ii) is a consequence from standard isoperimetric estimates which can be extracted from the proof of [KL2, Proposition 15]. Proposition 5.2 ([KL2]). Let G := (E, V ) be a graph and q : V [0, ). Then, for all U V we have on C c (U ). ( ) ( ) 1 1 αu 2 (deg +q) + q αu 2 (deg +q) Isoperimetric estimates at infinity. Let the Cheeger constant at infinity be defined as α = sup α V \K. K V finite Clearly, 0 α V α U α 1 for any U V. As a consequence of Proposition 5.2, we get the following theorem. Theorem 5.3. Let G := (E, V ) be a graph and q : V [0, ) be a potential. Assume α > 0. Then, we have the following: (a) For every ε > 0 there is k ε 0 such that on C c (V ) (1 ε) ( 1 1 α 2 ) (deg +q) kε + q (1 + ε) ( α 2 ) (deg +q) + kε. (b) D(( + q) 1/2 ) = D((deg +q) 1/2 ). (c) The operator + q has purely discrete spectrum if and only if we have lim inf x (deg + q)(x) =. In this case, if additionally α = 1, we get lim inf λ λ n ( + q) λ n (deg + q) = 1.

186 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 15 Proof. (a) Let ε > 0 and K V be finite and large enough such that (1 ε) ( 1 ) ( ) 1 α αv 2 \K ( ) ( ) αv 2 \K (1 + ε) α 2. From Proposition 5.2 we conclude on C c (V \ K ) (1 ε) ( 1 ) ( ) 1 α 2 (deg +q) 1 1 αv 2 \K (deg +q) + q ( ) αv 2 \K (deg +q) (1 + ε) ( α 2 ) (deg +q) By local finiteness the operators 1 V \K ( + q)1 V \K and 1 V \K (deg +q)1 V \K are bounded (indeed, finite rank) perturbations of + q and deg +q. This gives rise to the constants k ε and the inequality of (a) follows. Now, (b) is an immediate consequence of (a), and (c) follows by the Min-Max-Principle, Theorem A Relating sparseness and isoperimetric estimates. We now explain how the notions of sparseness and isoperimetric estimates are exactly related. First, we consider classical isoperimetric estimates. Theorem 5.4. Let G := (V, E ) be a graph, a, k 0, and let q : V [0, ) be a potential. (a) α V 1 if and only if (G, q) is (a, 0)-sparse. 1 + a (b) If (G, q) is (a, k)-sparse, then α V d k d(1 + a), where d := inf x V (deg +q)(x). In particular, α V > 0 if d > k. (c) Suppose that (G, q) is (a, k)-sparse graph that is not (a, k )-sparse for all k < k. Suppose also that there is d such that d = deg(x) + q(x) for all x V. Then α V = d k d(1 + a). Proof. Let W V be a finite set. Recalling the identity deg(w ) = 2 E W + W we notice that is equivalent to 1 W + q(w ) 1 + a (deg +q)(w ) 2 E W a( W + q(w )) which proves (a). For (b), the definition of (a, k)-sparseness yields W + q(w ) (deg +q)(w ) = 1 2 E W + q(w ) 1 a W (deg +q)(w ) (deg +q)(w ) k W (deg +q)(w ). This concludes immediately.

187 16 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER For (c), the lower bound of α V follows from (b). Since (G, q) is not (a, k )-sparse there is a finite W 0 V such that W 0 + q(w 0 ) (deg +q)(w 0 ) < 1 a W 0 + q(w 0 ) (deg +q)(w 0 ) k W 0 (deg +q)(w 0 ). Therefore α V < (d k )/(d(1 + a)). We next address the relation between almost sparseness and isoperimetry and show two almost equivalences. Theorem 5.5. Let G := (V, E ) be a graph and let q : V [0, ) be a potential. (a) If α > 0, then (G, q) is (a, k)-sparse for some a > 0, k 0. On the other hand, if (G, q) is (a, k)-sparse for some a > 0, k 0 and then l := lim inf(deg +q)(x) > k, x α l k l(a + 1) > 0, if l is finite and α 1/(1 + a) otherwise. (b) If α = 1, then (G, q) is almost sparse. On the other hand, if (G, q) is almost sparse and lim inf x (deg +q)(x) =, then α = 1. Proof. The first implication of (a) follows from Theorem 5.3 (a) and Theorem 2.2 (ii) (i). For the opposite direction let ε > 0 and K V be finite such that deg +q l ε on V \ K. Using the formula in the proof of Theorem 5.4 above, yields for W V \ K W + q(w ) (deg +q)(w ) = 1 2 E W (deg +q)(w ) 1 k (l ε) 1 k W + a( W + q(w )) (deg +q)(w ) a( W + q(w )). (deg +q)(w ) This proves (a). The first implication of (b) follows from Theorem 5.3 (a) and Theorem 3.2 (ii) (i). The other implication follows from (a) using the definition of almost sparseness. Remark 5.6. (a) We point out that without the assumptions on (deg +q) the converse implications do not hold. For example the Cayley graph of Z is 2-sparse (cf. Lemma 6.2), but has α = 0. (b) Observe that α = 1 implies lim inf x (deg +q)(x) =. Hence, (b) can be rephrased as the following equivalence: α = 1 is equivalent to (G, q) almost sparse and lim inf x (deg +q)(x) =. The previous theorems provides a slightly simplified proof of [K1] which also appeared morally in somewhat different forms in [D1, Woe]. Corollary 5.7. Let G := (V, E ) be a planar graph. (a) If for all vertices deg 7, then α V > 0. (b) If for all vertices away from a finite set deg 7, then α > 0. Proof. Combine Theorem 5.4 and Theorem 5.5 with Lemma 6.2.

188 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS Examples 6.1. Examples of sparse graphs. To start off, we exhibit two classes of sparse graphs. First we consider the case of graphs with bounded degree. Lemma 6.1. Let G := (V, E ) be a graph. Assume D := sup x V deg(x) < +, then G is D-sparse. Proof. Let W be a finite subset of V. Then, 2 E W deg(w ) D W. We turn to graphs which admit a 2-cell embedding into S g, where S g denotes a compact orientable topological surface of genus g. (The surface S g might be pictured as a sphere with g handles.) Admitting a 2-cell embedding means that the graphs can be embedded into S g without self-intersection. By definition we say that a graph is planar when g = 0. Note that unlike other possible definitions of planarity, we do not impose any local compactness on the embedding. Lemma 6.2. (a) Trees are 2-sparse. (b) Planar graphs are 6-sparse. (c) Graphs admitting a 2-cell embedding into S g with g 1 are 4g + 2-sparse. Proof. (a) Let G := (V, E ) be a tree and G W := (W, E W ) be a finite induced subgraph of G. Clearly E W W 1. Therefore, every tree is 2-sparse. We treat the cases (b) and (c) simultaneously. Let G := (V, E ) be a graph which is connected 2-cell embedded in S g with g 0 (as remarked above planar graphs correspond to g = 0). Let G W := (W, E W ) be a finite induced subgraph of G which, clearly, also admits a 2-cell embedding into S g. The statement is clear for W 2. Assume W 3. Let F W be the faces induced by G W := (W, E W ) in S g. Here, all faces (even the outer one) contain at least 3 edges, each edge belongs only to 2 faces, thus, 2 E W 3 F W. Euler s formula, W E W + F W = 2 2g, gives then 2 2g + E W = W + F W W E W that is This concludes the proof. E W 3 W + 6(g 1) max(2g + 1, 3) W. Next, we explain how to construct sparse graphs from existing sparse graphs. Lemma 6.3. Let G 1 := (V 1, E 1 ) and G 2 := (V 2, E 2 ) be two graphs. (a) Assume V 1 = V 2, G 1 is k 1 -sparse and G 2 is k 2 -sparse. Then, G := (V, E ) with E := max(e 1, E 2 ) is (k 1 + k 2 )-sparse. (b) Assume G 1 is k 1 -sparse and G 2 is k 2 -sparse. Then G 1 G 2 := (V, E ) with where V := V 1 V 2 and E ((x 1, x 2 ), (y 1, y 2 )) := δ {x1}(y 1 ) E 2 (x 2, y 2 ) + δ {x2}(y 2 ) E 1 (x 1, y 1 ), is (k 1 + k 2 )-sparse. (c) Assume V 1 = V 2, G 1 is k-sparse and E 2 E 1. Then, G 2 is k-sparse.

189 18 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER Proof. For (a) let W V be finite and note that E W E 1,W + E 2,W. For (b) let p 1, p 2 the canonical projections from V to V 1 and V 2. For finite W V we observe E W = E 1,p1(W ) + E 2,p2(W ) k 1 p 2 (W ) + k 2 p 1 (W ) (k 1 + k 2 ) W. For (c) and W V finite, we have E 2,W E 1,W which yields the statement. Remark 6.4. (a) We point out that there are bi-partite graphs which are not sparse. See for example [G, Proposition 4.11] or take an antitree, confer [KLW, Section 6], where the number of vertices in the spheres grows monotonously to. (b) The last point of the lemma states that the k-sparseness is non-decreasing when we remove edges from the graph. This is not the case for the isoperimetric constant Examples of almost-sparse and (a, k)-sparse graph. We construct a series of examples which are perturbations of a radial tree. They illustrate that sparseness, almost sparseness and (a, k)-sparseness are indeed different concepts. Let β = (β n ), γ = (γ n ) be two sequences of natural numbers. Let T = T (β) with T = (V, E T ) be a radial tree with root o and vertex degree β n at the n-th sphere, that is every vertex which has natural graph distance n to o has (β n 1) forward neighbors. We denote the distance spheres by S n. We let G (β, γ) be the set of graphs G := (V, E G ) that are super graphs of T such that the induced subgraphs G Sn are γ n -regular and E G (x, y) = E T (x, y) for x S n, y S m, m n. n Observe that G (β, γ) is non empty if and only if γ n j=0 (β j 1) is even and γ n < S n = n j=0 (β j 1) for all n 0. Figure 1. G with β = (3, 3, 4,...) and γ = (0, 2, 4, 5,...). Proposition 6.5. Let β, γ N N0 0, a = lim sup n γ n /β n and G G (β, γ). (a) If a = 0, then G is almost sparse. The graph G is sparse if and only if lim sup n γ n <. (b) If a > 0, then G is (a, k)-sparse for some k 0 if a > a. Conversely, if G is (a, k)-sparse for some k 0, then a a. Proof. Let ε > 0 and let N 0 be so large that γ n (a + ε)β n, n N.

190 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 19 Set C ε := N 1 n=0 degg (S n ). Let W be a non-empty finite subset of V. We calculate 2 E G W + G W = deg G (W ) = deg T (W ) + n 0 deg T (W ) + (a + ε) n 0 (1 + a + ε) deg T (W ) + C ε W W S n γ n N 1 W S n β n + W S n γ n n=0 = 2(1 + a + ε) E T W + (1 + a + ε) T W + C ε W (2(1 + a + ε) + C ε ) W + (1 + a + ε) T W, where we used that trees are 2-sparse in the last inequality. Finally, since G W T W, we conclude 2 E G W (2(1 + a + ε) + C ε ) W + (a + ε) G W. This shows that the graph in (a) with a = 0 is almost sparse and that the graph in (b) with a > 0 is (a + ε, k ε )-sparse for ε > 0 and k ε = 2(1 + a + ε) + C ε. Moreover, for the other statement of (a) let k 0 = lim sup n γ n and note that for G Sn 2 E Sn = γ n S n. Hence, if k 0 =, then G is not sparse. On the other hand, if k 0 <, then G is (k 0 + 2)-sparse by Lemma 6.3 as T is 2-sparse by Lemma 6.2. This finishes the proof of (a). Finally, assume that G is (a, k)-sparse with k 0. Then, for W = S n γ n S n = 2 E Sn k S n + a G S n = k S n + a β n S n Dividing by β n S n and taking the limit yields a a. This proves (b). Remark 6.6. In (a), we may suppose alternatively that we have the complete graph on S n and the following exponential growth lim n S n S n+1 = 0. Appendix A. Some general operator theory We collect some consequences of standard results from functional analysis that are used in the paper. Let H be a Hilbert space with norm. For a quadratic form Q, denote the form norm by Q := Q( ) + 2. The following is a direct consequence of the Closed Graph Theorem, (confer e.g. [We, Satz 4.7]). Theorem A.1. Let (Q 1, D(Q 1 )) and (Q 2, D(Q 2 )) be closed non-negative quadratic forms with a common form core D 0. Then, the following are equivalent: (i) D(Q 1 ) D(Q 2 ). (ii) There are constants c 1 > 0, c 2 0 such that c 1 Q 2 c 2 Q 1 on D 0. Proof. If (ii) holds, then any Q1 -Cauchy sequence is a Q2 -Cauchy sequence. Thus, (ii) implies (i). On the other hand, consider the identity map j : (D(Q 1 ), Q1 ) (D(Q 2 ), Q2 ). The map j is closed as it is defined on the whole Hilbert space (D(Q 1 ), Q1 ) and, thus, bounded by the Closed Graph Theorem [RS, Theorem III.12] which implies (i). For a selfadjoint operator T which is bounded from below, we denote the bottom of the spectrum by λ 0 (T ) and the bottom of the essential spectrum by λ ess 0 (T ). Let n(t ) N 0 { } be the dimension of the range of the spectral projection of

191 20 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER (, λ ess 0 (T )). For λ 0 (T ) < λ ess 0 (T ) we denote the eigenvalues below λ ess 0 (T ) by λ n (T ), for 0 n n(t ), in increasing order counted with multiplicity. Theorem A.2. Let (Q 1, D(Q 1 )) and (Q 2, D(Q 2 )) be closed non-negative quadratic forms with a common form core D 0 and let T 1 and T 2 be the corresponding selfadjoint operators. Assume there are constants c 1 > 0, c 2 R such that on D 0 c 1 Q 2 c 2 Q 1. Then, c 1 λ n (T 2 ) c 2 λ n (T 1 ), for 0 n min(n(t 1 ), n(t 2 )). Moreover, c 1 λ ess 0 (T 2 ) c 2 λ ess 0 (T 1 ), in particular, σ ess (T 1 ) = if σ ess (T 2 ) = and in this case Proof. Letting µ n (T ) = sup c 1 lim inf n ϕ 1,...,ϕ n H λ n (T 1 ) λ n (T 2 ). T ψ, ψ inf 0 ψ {ϕ 1,...,ϕ n} D 0 ψ, ψ, for a selfadjoint operator T, we know by the Min-Max-Principle [RS, Chapter XIII.1] µ n (T ) = λ n (T ) if λ n (T ) < λ ess 0 (T ) and µ n (T ) = λ ess 0 (T ) otherwise, n 0. Assume n min{n(t 1 ), n(t 2 )} and let ϕ (j) 0,..., ϕ(j) n be the eigenfunctions of T j to λ 0 (T j ),..., λ n (T j ) we get ( T 2 ψ, ψ ) c 1 λ n (T 2 ) c 3 = inf c 1 c 3 0 ψ {ϕ (2) 1,...,ϕ(2) n } D 0 ψ, ψ T 1 ψ, ψ inf µ n (T 1 ) = λ n (T 1 ) 0 ψ {ϕ (2) 1,...,ϕ(2) n } D 0 ψ, ψ This directly implies the first statement. By a similar argument the statement about the bottom of the essential spectrum follows, in particular, λ ess 0 (T 2 ) = implies lim n µ n (T 1 ) = and, thus, λ ess 0 (T 1 ) =. In this case λ n (T 2 ), n, which implies the final statement. Finally, we give a lemma which helps us to transform inequalities under form perturbations. Lemma A.3. Let (Q 1, D(Q 1 )), (Q 2, D(Q 2 )) and (q, D(q)) be closed symmetric non-negative quadratic forms with a common form core D 0 such that there are α (0, 1), C α 0 such that on D 0. If for a (0, 1) and k 0 then q αq 1 + C (1 a)q 2 k Q 1 on D 0, (1 α)(1 a) (1 α(1 a)) (Q 2 q) (1 α)k + ac α (1 α(1 a)) Q 1 q, on D 0. In particular, if a 0 +, then (1 α)(1 a)/(1 α(1 a)) 1 and if α 0 +, then (1 α)(1 a)/(1 α(1 a)) (1 a).

192 EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 21 Proof. The assumption on q implies α q (1 α) (Q 1 q) + C α (1 α). We subtract (1 a)q on each side of the lower bound in (1 a)q 2 k Q 1. Then, we get (1 a)(q 2 q) k (Q 1 q) + aq and, thus, the asserted inequality follows. 1 α(1 a) (Q 1 q) + ac α (1 α) (1 α) Acknowledgement. MB was partially supported by the ANR project HAB (ANR-12-BS ). SG was partially supported by the ANR project GeRaSic and SQFT. MK enjoyed the hospitality of Bordeaux University when this work started. Moreover, MK acknowledges the financial support of the German Science Foundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (grant no. 1105/10 and no. 225/10) and BSF grant no References [AABL] N. Alon, O. Angel, I. Benjamini, E. Lubetzky, Sums and products along sparse graphs, Israel J. Math. 188 (2012), [BHJ] F. Bauer, B. Hua, J. Jost, The dual Cheeger constant and spectra of infinite graphs, Adv. in Math., 251, 30 (2014), [BJL] F. Bauer, J. Jost, S. Liu, Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator, Mathematical research letters, 19 (2012) 6, [BKW] F. Bauer, M. Keller, R.K. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, to appear in J. Eur. Math. Soc. (JEMS), arxiv: , (2012). [B] J. Breuer, Singular continuous spectrum for the Laplacian on certain sparse trees, Comm. Math. Phys. 269, no. 3, (2007) [D1] J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284, no. 2, (1984) [D2] J. Dodziuk, Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, 2006, [DK] J. Dodziuk, W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, From Local Times to Global Geometry, Control and Physics, Pitman Res. Notes Math. Ser., 150, (1986) [DM] J. Dodziuk, V. Matthai, Kato s inequality and asymptotic spectral properties for discrete magnetic Laplacians, The ubiquitous heat kernel, Cont. Math. 398, Am. Math. Soc. (2006), [EGS] P. Erdös, R.L. Graham, E. Szemerédi, On sparse graphs with dense long paths, Computers and Mathematics with Applications 1, Issues 3-4, (1975) [F] K. Fujiwara, Laplacians on rapidly branching trees, Duke Math Jour. 83, (1996) [G] S. Golénia, Hardy inequality and eigenvalue asymptotic for discrete Laplacians, to appear in J. Funct. Anal., arxiv: [GG] V. Georgescu, S. Golénia, Decay Preserving Operators and stability of the essential spectrum, J. Operator Theory 59, no. 1, (2008) [GKS] B. Güneysu, M. Keller, M. Schmidt, A Feynman-Kac-Itō Formula for magnetic Schrödinger operators on graphs, 2012, arxiv: [H] Y. Higuchi, Combinatorial Curvature for Planar Graphs, Journal of Graph Theory 38, Issue 4, (2001) [JL] J. Jost, S. Liu, Ollivier s Ricci curvature, local clustering and curvature dimension inequalities on graphs, preprint 2011, arxiv: v2. [K1] M. Keller, Essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346, (2010) [K2] M. Keller, Curvature, geometry and spectral properties of planar graphs, Discrete Comput. Geom., 46, (2011),

193 22 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER [KL1] M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math. 666, (2012), [KL2] M. Keller, D. Lenz, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5, no. 4, (2010) [KLW] M. Keller, D. Lenz, R.K. Wojciechowski, Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs, Math. Z. 274, Issue 3 (2013), [KP] M. Keller, N. Peyerimhoff, Cheeger constants, growth and spectrum of locally tessellating planar graphs, Math. Z., 268, (2011), [LS] A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (2008), no. 8, [LY] Y. Lin, S.T. Yau, Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett. 17 (2010), [L] M. Loréa, On matroidal families, Discrete Math. 28, (1979) [M1] B. Mohar, Isoperimetics inequalities, growth and the spectrum of graphs, Linear Algebra Appl. 103, (1988), [M2] B. Mohar, Some relations between analytic and geometric properties of infinite graphs, Discrete Math. 95 (1991), no. 1 3, [M3] B. Mohar, Many large eigenvalues in sparse graphs, European J. Combin. 34, no. 7, (2013) [RS] M. Reed, B. Simon, Methods of Modern Mathematical Physics I, II, IV: Functional analysis. Fourier analysis, Self-adjointness, Academic Press, New York e.a., [SV] P. Stollmann, J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), [We] J. Weidmann, Lineare Operatoren in Hilberträumen 1, B. G. Teubner, Stuttgart, [Woe] W. Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Camb. Phil. Soc. 124, (1998) [Woj1] R. K. Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008, Thesis (Ph.D.) City University of New York. [Woj2] R. K. Wojciechowski,Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J. 58 (2009), Michel Bonnefont, Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F Talence cedex, France address: michel.bonnefont@math.u-bordeaux1.fr Sylvain Golénia, Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F Talence cedex, France address: sylvain.golenia@math.u-bordeaux1.fr Matthias Keller, Friedrich Schiller Universität Jena, Mathematisches Institut, Jena, Germany address: m.keller@uni-jena.de

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195 CHAPTER 8 B. Hua, M. Keller, Harmonic functions of general graph Laplacians, Calculus of Variations and Partial Differential Equations 51 (2014),

196 Calc. Var. (2014) 51: DOI /s Calculus of Variations Harmonic functions of general graph Laplacians Bobo Hua Matthias Keller Received: 10 April 2013 / Accepted: 5 September 2013 / Published online: 1 October 2013 Springer-Verlag Berlin Heidelberg 2013 Abstract We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an L p Liouville type theorem which is a quantitative integral L p estimate of harmonic functions analogous to Karp s theorem for Riemannian manifolds. As corollaries we obtain Yau s L p -Liouville type theorem on graphs, identify the domain of the generator of the semigroup on L p and get a criterion for recurrence. As a side product, we show an analogue of Yau s L p Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces. Mathematics Subject Classification 31C05 58E20 1 Introduction The study of harmonic functions is a fundamental topic in various areas of mathematics. An important question is which subspaces of harmonic functions are trivial, that is, they contain only constant functions. Such results are referred to as Liouville type theorems. In Riemannian geometry L p -Liouville type theorems for harmonic functions were studied for example by Yau [58], Karp [32], Li Schoen [41] and many others. Karp s criterion was later generalized by Sturm [49] to the setting of strongly local regular Dirichlet forms. Over the years there were several attempts to realize an analogous theorem for graphs, see Holopainen Soardi [25], Rigoli Salvatori Vignati [46], Masamune [44] Communicated by J. Jost. B. Hua (B) Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany bobohua@mis.mpg.de M. Keller Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel mkeller@ma.huji.ac.il 123

197 344 B. Hua, M. Keller and most recently Hua Jost [20]. In all these works normalized Laplacians were studied (often with further restrictions on the vertex degree) and certain criteria, all weaker than Karp s integral estimate, were obtained. The main challenge when considering graphs is the non-existence of a chain rule and, moreover, the fact that for unbounded graph Laplacians the natural graph distance is very often not the proper analogue to the Riemannian distance in manifolds. In this paper, we use the newly developed concept of intrinsic metrics on graphs to prove an analogue to Karp s theorem for general Laplacians on weighted graphs. Thus, we generalize all earlier results on graphs not only with respect to the generality of the setting but also by recovering the precise analogue of Karp s criterion. Harmonic maps are very important nonlinear objects in geometric analysis studied thoroughly by many authors (e.g. Eells Sampson [11], Schoen Yau [53], Hildebrandt Jost Widman [21]). In this paper, we adopt a definition of harmonic maps between metric measure spaces introduced by Jost [27 30]. In particular, we study harmonic maps from graphs into Hadamard spaces (i.e. globally non-positively curved spaces, also called CAT(0)-spaces), studied also by [26,31,39], and prove Liouville type theorems in this context. For various Liouville theorems on manifolds, we refer to [6,9,34,54] and references therein. We prove the finite-energy Liouville theorem for harmonic maps from graphs into Hadamard spaces analogous to the one in Cheng Tam Wang [9] on manifolds. In what follows we first state and discuss our results and refer for details and precise definitions to Sect. 2. Our framework are weighted graphs over a discrete measure space (X, m) introduced in [35] which includes non locally finite graphs, (see also [48]). In this setting a pseudo metric is called intrinsic if the energy measures of distance functions can be estimated by the measure of the graph (see Definition 2.2). We further call such a pseudo metric compatible if it has finite jump size and the weighted vertex degree is bounded on each distance ball (see Definition 2.3). As the boundedness of the weighted vertex degree is implied by finiteness of distance balls which is equivalent to metric completeness in the case of a path metric on a locally finite graph, see [24, Theorem A.1], this assumption can be seen as an analogue of completeness in the Riemannian manifold case. Similarly, Sturm [49] asks for precompactness of balls. Our first main result is the following analogue to Karp s L p Liouville theorem [32, Theorem 2.2], whose proof is given in Sect A function is called (sub)harmonic if it is in the domain of the formal Laplacian and the formal Laplacian applied to this function is pointwise (less than or) equal to zero, (see Definition 2.1). We denote by 1 Br the characteristic function of the balls B r, r 0, which are taken with respect to an intrinsic metric about a fixed vertex o X. Theorem 1.1 (Karp s L p Liouville theorem) Assume a connected weighted graph allows for a compatible intrinsic metric. Then every non-negative subharmonic function f satisfying inf r 0 >0 r 0 r f 1 Br p dr =, p for some p (1, ), is constant. Clearly, the integral in the theorem above diverges, whenever 0 = f L p (X, m). Thus, as an immediate corollary, we get Yau s L p Liouville type theorem [58]. Corollary 1.2 (Yau s L p Liouville theorem) Assume a connected weighted graph X allows for a compatible intrinsic metric. Then every non-negative subharmonic function in L p (X, m), p (1, ), is constant. 123

198 Harmonic functions of general graph Laplacians 345 Remark 1.3 (a) The results above imply the corresponding statements for harmonic functions by the simple observation that f +, f and f of a harmonic function f are nonnegative and subharmonic. (b) Harmonicity of a function is independent of the choice of the measure m. Hence, for any non-constant harmonic function f on X, we may find a sufficiently small measure m such that f L p (X, m) for any p (0, ), see[44]. Our theorem states that if we impose the restriction of compatibility on the measure and the metric, then the L p Liouville theorem holds for 1 < p <. (c) Theorem 1.1 generalizes all earlier results on graphs [20,25,44,46] for the case p (1, ). Not only that our setting is more general as the natural graph distance is always a compatible intrinsic metric to the normalized Laplacian but also our criterion is more general. In particular, if f satisfies lim sup r 1 r 2 log r f 1 B r p p <, then the integral in Theorem 1.1 diverges. Thus, Theorem 1.1 is stronger than [20, Theorem 1.1] (which had only r 2 rather than r 2 log r in the denominator). The authors of [20] observed that for the normalized Laplacian the case p (1, 2] can already be obtained by their techniques, (see [20, Remark 3.3]). Here, the missing cases p (2, ) are treated by adopting a subtle lemma in [25]. Moreover, our techniques would also allow a statement such as [20, Theorem 1.1] for the cases p < 1. (d) In [35] discrete measure spaces (X, m) with the assumption that every infinite path has infinite measure are discussed (this assumption is denoted by (A) in [35]). It is not hard to see that for connected graphs over (X, m) every non-negative subharmonic function L p (X, m), p [1, ), is trivial. In fact, from every non-constant positive subharmonic function we can extract a sequence of vertices such that the function values increase along this sequence (compare [35, Lemma 3.2 and Theorem 8]). Since this path has infinite measure, the function is not contained in L p (X, m), p [1, ). Thus, the only interesting measure spaces are those that contain an infinite path of finite measure. (e) Sturm [49] proves an analogue for Karp s theorem for weakly subharmonic functions. This might seem stronger, however, in our setting on graphs weak solutions of equations are automatically solutions, [23, Theorem 2.2 and Corollary 2.3]. Corollary 1.2 allows us to explicitly determine the domain of the generator L p of the semigroup on L p (X, m). We denote by the formal Laplacian with formal domain F.(For definitions see Sect. 2.2). The proof of the corollary below is given in Sect Corollary 1.4 (Domain of the L p generators) Assume a connected weighted graph X allows for a compatible intrinsic metric. Then, for p (1, ), the generator L p is a restriction of and D(L p ) ={u L p (X, m) F u L p (X, m)}. Remark 1.5 (a) The corollary above generalizes [24, Theorem 1] to the case p (1, ) and settles the question in [24, Remark 3.6]. Moreover, it complements [35, Theorem5]. (b) It would be interesting to know whether there is a Liouville type theorem for functions in D(L p ) without the assumption of compatibility on the metric. We get furthermore a sufficient criterion for recurrence analogous to [32, Theorem 3.5] and [49, Theorem 3] which generalizes for example [10, Theorem 2.2], [46, Corollary B], 123

199 346 B. Hua, M. Keller [55, Lemma 3.12], [18, Corollary 1.4], [45, Theorem 1.2] on graphs. For a characterization of recurrence see Proposition 3.3 in Sect. 3.4, where also the proof of the corollary below is given. Corollary 1.6 (Recurrence) Assume a connected weighted graph allows for a compatible intrinsic metric. If then the graph is recurrent. 1 r dr =, m(b r ) Contrary to the normalized Laplacian, [20, Theorem 1.2], there is no L 1 Liouville type theorem in the general case. However, for stochastic complete graphs (see Sect. 4) wehave the following analogue to [16, Theorem 3], [49, Theorem 2]. The proof following [17] is given in Sect. 4. We also give counter-examples to L 1 Liouville theorem which complement the counter-examples from manifolds, [8,41]. Theorem 1.7 (Grigor yan s L 1 theorem) Assume a connected graph X is stochastically complete. Then, every non-negative superharmonic function in L 1 (X, m) is constant. For vertices x, y X that are connected by an edge, we denote a directed edge by xy and the positive symmetric edge weight by μ xy.wedefine xy f = f (x) f (y). The following L p Caccioppoli-type inequality is a side product of our analysis. Such an inequality was proven in [20,25,46] for bounded operators. The classical Caccioppoli inequality is the case p = 2, which can be found for graphs in [5,24,43]. Theorem 1.8 (Caccioppoli-type inequality) Assume a connected weighted graph allows for a compatible intrinsic metric and p (1, ). Then, there is C > 0 such that for every non-negative subharmonic function f and all 0 < r < R 3s μ xy ( f (x) f (y)) p 2 xy f 2 C f 1 (R r) 2 BR \B r p p, x,y B r where s is the jump size of the intrinsic metric (see Sect. 2.3). Remark 1.9 (a) The theorem above allows for a direct proof of Corollary 1.2, confer [20, Corollary 3.1]. (b) For p 2, we can strengthen the inequality by replacing ( f (x) f (y)) p 2 on the left hand side by f p 2 (x) + f p 2 (y), see Remark 3.2 in Sect. 3.3, where the theorem is proven. The following quantitative consequence of Theorem 1.1 which is a generalization of Corollary 1.2 has various corollaries that are stated and proven in Sect. 5. For an intrinsic metric ρ and a fixed vertex o X let 123 ρ 1 = 1 ρ(, o).

200 Harmonic functions of general graph Laplacians 347 Theorem 1.10 Assume a connected weighted graph X allows for a compatible intrinsic metric ρ. If a non-negative subharmonic function f satisfies f L p (X, mρ1 2 ), for some p (1, ), then f is constant. Next, we turn to harmonic maps from graphs into Hadamard spaces, (see Sect. 6, in particular Definition 6.1). We prove the following consequence of Karp s theorem in Sect. 6. Theorem 1.11 (Karp s theorem for harmonic maps) Assume a connected weighted graph X allows for a compatible intrinsic metric ρ. Let u be a harmonic map into an Hadamard space (Y, d). If there are p (1, ) and y Y such that d(u( ), y) L p (X, mρ1 2 ), then u is bounded. Moreover, if mρ1 2 (X) = or y is in the image of u, then u is constant. Finally, we turn to harmonic functions and maps of finite energy, (for definitions see Sects. 2.2 and 6.2). The two theorems below stand in close relationship to the celebrated theoremofkendall[33, Theorem 6], (confer [22,40]). Our first result in this line is a direct consequence of Theorem 6.3 and it is an analogue to Cheng Tam Wang [9, Theorem 3.1]. Theorem 1.12 Assume that on a graph every harmonic function of finite energy is bounded. Then, every harmonic map from the graph into an Hadamard space is bounded. The second result in this line is an analogue to [9, Theorem 3.2]. An Hadamard space is called locally compact if for any point there exists a precompact neighborhood. Theorem 1.13 Assume that on a graph every bounded harmonic function is constant. Then, every finite-energy harmonic map from the graph into a locally compact Hadamard space is constant. The paper is organized as follows. In the next section, we introduce the involved concepts and recall some basic inequalities. Section 3 is devoted to the proofs of Theorems 1.1, 1.8 and the corollaries above. The proof of Theorem 1.7 and counter-examples to an L 1 -Liouville type statement are given in Sect. 4. In Sect. 5 we prove Theorem 1.10 and derive various corollaries. Harmonic maps from graphs into Hadamard spaces are discussed in Sect. 6. Theorem 1.11 isproveninsect.6.1 and Theorems 1.12 and 1.13 are proven in Sect Several applications are discussed in Sect Throughout this paper C always denotes a constant that might change from line to line. Moreover, we use the convention that 0 = 0, (which only appears in expressions such as f q (x) xy f with f (x) = f (y) = 0andq > 0). 2 Set-up and preliminaries 2.1 Weighted graphs Let X be a countable discrete set and m : X (0, ). Extending m additively to sets, (X, m) becomes a measure space with a measure of full support. A graph over (X, m) is 123

201 348 B. Hua, M. Keller induced by an edge weight function μ : X X [0, ), (x, y) μ xy that is symmetric, has zero diagonal and satisfies μ xy <, x X. y X If μ xy > 0 we write x y and let xy and yx be the oriented edges of the graph. We write xy A for a set A X if both of the vertices of the edge xy are contained in A, i.e., x, y A. When we fix an orientation for the edges we denote the directed edges often by e. We refer to the triple (X,μ,m) as a weighted graph. Weassumethegraphisconnected, that is for every two vertices x, y X there is a path x = x 0 x 1 x n = y. The spaces L p (X, m), p [1, ), and L (X) are defined in the natural way. For p [1, ), letp be its Hölder dual, i.e., 1 p + 1 p = Laplacians and (sub)harmonic functions We define the formal Laplacian on the formal domain F(X) = f : X R μ xy f (y) < for all x X, y X by f (x) = 1 μ xy ( f (x) f (y)). m(x) y X Definition 2.1 (Harmonic function) A function f : X R is called harmonic (subharmonic, superharmonic)if f F(X) and f = 0, ( f 0, f 0). Obviously, the measure does not play a role in the definition of harmonicity. We denote by L the positive selfadjoint restriction of on L 2 (X, m) which arises from the closure Q of the restriction of the quadratic form E :{X R} [0, ] E( f ) = 1 2 x,y X μ xy xy f 2 to C c (X), the space of finitely supported functions, (for details see [35]). Since Q is a Dirichlet form, the semigroup e tl, t 0, extends to a C 0 -semigroup on L p (X, m), p [1, ) (resp. a weak C 0 -semigroup for p = ). We denote the generators of these semigroups by L p, p [1, ). Moreover, we say a function f has finite energy if E( f )<. 2.3 Intrinsic metrics Next, we introduce the concept of intrinsic metrics. A pseudo metric is a symmetric map X X [0, ) with zero diagonal which satisfies the triangle inequality. Definition 2.2 (Intrinsic metric) A pseudo metric ρ on X is called an intrinsic metric if μ xy ρ 2 (x, y) m(x), x X. 123 y X

202 Harmonic functions of general graph Laplacians 349 If for a function f : X R the map Ɣ( f ) : x y X μ xy xy f 2 takes finite values, then Ɣ( f ) defines the energy measure of f. Thus, a pseudo metric ρ is intrinsic if the energy measures Ɣ(ρ(x, )), x X, are absolutely continuous with respect to m with Radon-Nikodym derivative dm d Ɣ(ρ(x, )) = Ɣ(ρ(x, ))/m satisfying Ɣ(ρ(x, ))/m 1. In various situations the natural graph distance proves to be insufficient for the investigations of unbounded Laplacians, see [36,56,57]. For this reason the concept of intrinsic metrics developed in [12] for regular Dirichlet forms received quite some attention as a candidate to overcome these problems. Indeed, intrinsic metrics already have been applied successfully to various problems on graphs [3,4,13,14,24] and related settings [15]. The jumps size s of a pseudo metric is given by s := sup{ρ(x, y) x, y X, x y} [0, ]. From now on, ρ always denotes an intrinsic metric and s denotes its jump size. We fix a base point o X which we suppress in notation and denote the distance balls by B r ={x X ρ(x, o) r}, r 0. Since ρ takes values in [0, ) in our setting, the results are indeed independent of the choice of o.foru X, we write B r (U) ={x X ρ(x, y) r for some y U}, r 0. Define the weighted vertex degree Deg : X [0, ) by Deg(x) = 1 μ xy, x X. m(x) y X Definition 2.3 (Compatible metric) A pseudo metric on X is called compatible if it has finite jump size and the restriction of Deg to every distance ball is bounded, i.e., Deg Br C(r) < for all r 0. Example 2.4 (a) For any given weighted graph there is an intrinsic path metric defined by δ(x, y) = n 1 inf x=x 0 x n =y i=0 (Deg(x i ) Deg(x i+1 )) 1 2. This intrinsic metric can be turned into an intrinsic metric δ r with finite jump size s = r by taking the path metric with edge weights δ(x, y) r, x y. In many cases, neither δ r nor δ is compatible. (b) If the measure m is larger than the measure n(x) = y X μ xy, x X, then the natural graph distance (i.e., the path metric with edge weights 1) is an intrinsic metric which is compatible since s = 1andDeg 1 in this case. Remark 2.5 (a) In view of Example 2.4 (b) it is apparent that [20, Theorem 1.1] is included in Theorem 1.1. (b) In [24, Theorem A.1] a Hopf-Rinow type theorem is shown which states that for a locally finite graph a path metric is complete if and only if all balls are finite. Thus, compatibility can be seen as a completeness assumption of the graph. (c) It is not hard to see that there are graphs that do not allow for a compatible intrinsic metric. However, to a given edge weight function μ and a pseudo metric ρ, we can always assign a minimal measure m such that ρ is intrinsic, i.e., let m(x) = y X μ xyρ 2 (x, y), x X. If ρ already has finite jump size and all balls are finite, then ρ is automatically compatible. (d) The assumption that Deg is bounded on distance balls is equivalent to either of the following assumptions 123

203 350 B. Hua, M. Keller (i) The restriction of to any distance ball (with Dirichlet boundary conditions) is a bounded operator. (ii) The Radon-Nikodym derivative of the measure n given by n(x) = y μ xy, x X, with respect to the measure m is bounded on the distance balls. The equivalence of (i) follows from Theorem [23, Theorem 9.3] and the one of (ii) is obvious. In the subsequent, we will make use of the cut-off function η = η r,r,0 r < R, onx given by ( ) R ρ(, o) η = 1. R r Lemma 2.6 Let η = η r,r, 0 < r < R, be given as above. Then, (a) η Br 1 and η X\BR 0. (b) For x X, μ xy xy η 2 y X + 1 (R r) 2 1 B R+s \B r s (x)m(x). Proof (a) is obvious from the definition of η and (b) follows directly from xy η 1 R r ρ(x, y)1 B R+s \B r s (x) for x y and the intrinsic metric property of ρ. 2.4 Green s formula, Leibniz rules and mean value theorem We first prove a Green s formula which is an L p version of the one in [24]. Lemma 2.7 (Green s formula) Let p [1, ),U X and assume Deg is bounded on U. Then for all f with f 1 U L p (X, m) F(X) and g L p (X, m) with B s (supp g) U ( f )(x)g(x)m(x) = 1 μ xy xy f xy g. 2 x X x,y U Proof The formal calculation in the proof of Green s formula is a straightforward algebraic manipulation. To ensure that all involved terms converge absolutely, one invokes Hölder s inequality and the boundedness assumption on Deg (confer the proof of Lemma 3.1 and 3.3 in [24]). The following Leibniz rules follow by direct computations. Lemma 2.8 (Leibniz rules) For all x, y X, x y and f, g : X R xy ( fg) = f (y) xy g + g(x) xy f = f (y) xy g + g(y) xy f + xy f xy g. A fundamental difference of Laplacians on graphs and on manifolds is the absence of a chain rule in the graph case. In particular, existence of a chain rule can be used as a characterization for a regular Dirichlet form to be strongly local. We circumvent this problem by using the mean value theorem from calculus instead. In particular, for a continuously differentiable function φ : R R and f : X R, wehave 123 xy (φ f ) = φ (ζ ) xy f, for some ζ [f (x) f (y), f (x) f (y)].

204 Harmonic functions of general graph Laplacians 351 In this paper we will apply this formula to get estimates for the function φ : t t p 1, p (1, ). However, we need a refined inequality as it was already used in the proof of [25, Theorem 2.1]. For the convenience of the reader, we include a short proof here. Lemma 2.9 (Mean value inequalities) For all f : X R and x y with xy f 0, (a) xy f p ( f p 2 (x) + f p 2 (y)) xy f,for p [2, ), (b) xy f p 1 C( f (x) f (y)) p 2 xy f,for p (1, ), wherec= (p 1) 1. Proof (a) Denote a = f (y), b = f (x). As it is the only non-trivial case, we assume 0 < a < b. Note that for p = 1 b p 1 a p 1 = (b a)(b p 2 + a p 2 ) + ab(b p 3 a p 3 ). Thus, the statement is immediate for p 3 since the second term on the right side is nonnegative in this case. Let 2 p < 3 and note a p 3 > b p 3. The function t t 2 p is convex on (0, ) and, thus, its image lies below the line segment connecting (b 1, b p 2 ) and (a 1, a p 2 ). Therefore, a p 3 b p 3 a p 3 b p 3 (3 p) = a 1 b 1 t ( (b 2 p dt (a 1 b 1 p 2 a p 2 ) ) ) +a p 2 2 = 1 2ab (b a)(a p 2 + b p 2 ). From the equality in the beginning of the proof we now deduce the assertion in the case 2 p < 3. (b) The case p 2 follows from (a). The case 1 < p 2 in (b) follows directly from the mean value theorem. 3 Proofs for harmonic functions In this section we prove the main theorems and the corresponding corollaries for harmonic functions. It will be convenient to introduce the following orientation on the edges. For a given non-negative subharmonic function f,welete f be the set of oriented edges e = e + e such that 3.1 The key estimate e f 0, i.e., f (e + ) f (e ). The lemma below is vital for the proof of Theorems 1.1 and 1.8. Lemma 3.1 Let p (1, ), 0 ϕ L (X) and U = B s (supp ϕ). Assume Deg is bounded on U. Then, for every non-negative subharmonic function f with f 1 U L p (X, m), e E f μ e f p 2 (e + )ϕ 2 (e ) e f 2 C e E f,e U μ e f p 1 (e + )ϕ(e ) e f e ϕ, where C = 2/((p 1) 1). 123

205 352 B. Hua, M. Keller Proof From the assumptions f 1 U L p (X, m) and ϕ L (X), weinferϕ 2 f p 1 L p (X, m) (as p = p/(p 1)). Thus, compatibility of the pseudo metric implies applicability of Green s formula with f and g = ϕ 2 f p 1. We start by using non-negativity and subharmonicity of f before applying Green s formula (Lemma 2.7) and thefirst and second Leibniz rule (Lemma 2.8) 0 ( f )(x)(ϕ 2 f p 1 )(x)m(x) = x X = e U = e U C e U e E f,e U μ e e f [ ϕ 2 (e ) e f p 1 + f p 1 (e + ) e ϕ 2] μ e e f e (ϕ 2 f p 1 ) μ e e f [ ϕ 2 (e ) e f p f p 1 (e + )ϕ(e ) e ϕ + f p 1 (e + ) e ϕ 2] μ e f p 2 (e + )ϕ 2 (e ) e f e U μ e f p 1 (e + )ϕ(e ) e f e ϕ, where we dropped the third term in the third line since it is non-negative because of e f 0 and we estimated the first term on the right hand side using the mean value theorem, Lemma 2.9 (b). Absolute convergence of the two terms in the last line can be checked using Hölder s inequality and the assumptions f 1 U L p (X, m), ϕ L (X) and boundedness of Deg on U. Hence, we obtain the statement of the lemma. 3.2 Proof of Karp s theorem Proof of Theorem 1.1 Let p (1, ) and let f be a non-negative subharmonic function. Assume f 1 Br L p (X, m) for all r 0 since otherwise inf r0 r 0 r/ f 1 Br p pdr = 0. Let η = η r+s,r s with 0 < r < R 3s (see Sect. 2.3). Then by Lemma 3.1 (applied with ϕ = η) we obtain (noting additionally that xy η = 0, x, y B r ) μ e f p 2 (e + )η 2 (e ) e f 2 C e B R \B r μ e f p 1 (e + )η(e ) e f e η. e B R Now, the Cauchy-Schwarz inequality, e μ e f p (e + ) e η 2 x,y μ xy f p (x) xy η 2 and the cut-off function lemma, Lemma 2.6, yield 2 μ e f p 2 (e + )η 2 (e ) e f 2 e B R C μ e f p (e + ) e η 2 f p 2 (e + )η 2 (e ) e f 2 e B R \B r e B R \B r C (R r) 2 f 1 B R \B r p p f p 2 (e + )η 2 (e ) e f 2. e B R Let R 0 3s be such that f 1 BR0 = 0 and denote 123 e B r v(r) = f 1 Br p p, r 0.

206 Harmonic functions of general graph Laplacians 353 Moreover, for j 0, let R j = 2 j R 0, ϕ j = η R j +s,r j+1 s and Q j+1 = μ e f p 2 (e + )ϕ 2 j (e ) e f 2. e B R j+1 As ϕ j 1 ϕ j,wegetq j Q j+1 and together with the estimate above this implies Q j Q j+1 Q 2 j+1 C v(r j+1 ) (R j+1 R j ) 2 (Q j+1 Q j ), j 0. Since R j+1 = 2R j, dividing the above inequality by v(r j+1) Q R 2 j Q j+1 and adding C/Q j+1 j+1 yield and, thus, R 2 j+1 v(r j+1 ) + C Q j+1 C Q j 1 C j=1 R 2 j+1 v(r j+1 ) 1 Q 1. Now, the assumption R 0 r/v(r)dr = implies R 2 j j=0 v(r j ) =. Therefore, Q 1 = 0. As this is true for all R 0 large enough, we have f p 2 (e + ) e f 2 = 0, for all edges e. Forp 2, connectedness clearly implies that f is constant. On the other hand, for p (1, 2], wealwayshave f p 2 (e + )>0and, thus, f is constant. 3.3 Proof of the Caccioppoli inequality Proof of Theorem 1.8 Using Lemma 3.1 and the inequality ab εa 2 + 4ε 1 b2, ε>0, we estimate μ e f p 2 (e + )ϕ 2 (e ) e f 2 C μ e f p 1 (e + )ϕ(e ) e f e ϕ e E f e E f 1 μ e f p 2 (e + )ϕ 2 (e ) e f 2 + C μ e f p (e + ) e ϕ 2. 2 e E f e E f Letting ϕ = η = η r+s,r s with 0 < r < R 3s (from Sect. 2.3) and using the cut-off function lemma, Lemma 2.6, we arrive at μ e f p 2 (e + ) e f 2 C e E f μ e f p (e + ) e η 2 C f 1 (R r) 2 BR \B r p p. e E f Remark 3.2 In order to obtain the stronger statement for p [2, ) mentioned in Remark 1.9 (b), we invoke Lemma 2.9 (a) in the proof of Lemma 3.1 instead of Lemma 2.9 (b) and proceed as in the proof above. 123

207 354 B. Hua, M. Keller 3.4 Proof of the corollaries In this section we prove the corollaries. Proof of Corollary 1.2 (Yau s L p Liouville theorem) Clearly the integral in Theorem 1.1 diverges if f L p (X, m). Proof of Corollary 1.4 (Domain of the L p generators) Let f L p (X, m) F(X) be such that ( + 1) f = 0. Since the positive and negative part f +, f of f are non-negative, subharmonic and in L p (X, m), they must be constant by Corollary 1.2. This implies f ± 0 and, thus, f 0. Now, the proof of the corollary works literally line by line as the proof of [35, Theorem 5]. For the proof of Corollary 1.6 we recall the following well known equivalent conditions for recurrence. Proposition 3.3 (Characterization of recurrence) Let a connected graph X be given. Then the following are equivalent. (i) For the transition matrix P with P x,y = μ xy / z X μ xz, x, y X, we have n=0 P (n) (x, y) = for some (all) x, y X, where P (n) denotes the nth power of P. (ii) For m 1 and some (all) x, y X, we have 0 e tl δ x (y)dt =,whereδ x (y) = 1 if x = y and zero otherwise. (iii) For all m and some (all) x, y X, we have 0 e tl δ x (y)dt =. (iv) Every bounded superharmonic (or subharmonic) function is constant. (v) Every non-negative superharmonic function is constant. (vi) Every superharmonic (or subharmonic) function of finite energy is constant. (vii) cap(x) := inf{e( f ) f C c (X), f (x) = 1} =0 for some (all) x X A graph is called recurrent if one of the equivalent statements of Proposition 3.3 is satisfied. Proof The equivalence (i) (ii) is shown in [47, Theorem 6] (confer [7, Theorem 4.34]). The equivalences (ii) (vi) (iii) are in [47, Theorem 2 and Theorem 9] (confer [48, Theorem 3.34]). The equivalences (i) (v) (vii) are found in [55, Theorem 1.16,Theorem 2.12]. The equivalence (iv) (v) follows since every non-negative superharmonic function f can be approximated by the bounded superharmonic functions f n, n 1. Proof of Corollary 1.6 (Recurrence) Theorem 1.1 implies that any non-negative bounded subharmonic function f is constant provided inf r0 r 0 r/m(b r )dr = since f 1 Br p p f m(b p r ), r 0. By Proposition 3.3 the graph is recurrent. 4 L 1 -Liouville theorem and counter-examples In this section we deal with the borderline case of the L p Liouville theorem p = 1. We first prove Theorem 1.7 which deals with the stochastic complete case and then give two examples which show that there is no L 1 Liouville theorem for non-negative subharmonic functions in the general case. A graph is called stochastically complete if e tl 1 = 1, where 1 denotes the function that is constantly one on X. For the relevance of the concept see [17,35,56]. The proof of Theorem 1.7 follows along the lines of the proof of [17, Theorem 13.2]. 123

208 Harmonic functions of general graph Laplacians 355 Proof of Theorem 1.7 If the graph is recurrent, then there are no non-constant non-negative superharmonic functions by Proposition 3.3. So assume the graph is not recurrent which implies G(x, y) = 0 e tl δ x (y)dt <, x, y X, again by Proposition 3.3. LetK n, n 0, be an sequence of finite sets exhausting X and G n (x, y) = 0 e tl n δ x (y)dt, where L n are the finite dimensional operators arising from the restriction of the form Q to C c (K n ). By domain monotonicity, [35, Proposition 2.6 and 2.7] the semigroups e tl n converge monotonously increasing to e tl and, hence, G n (x, y) G(x, y) for x, y K n, and G n G, n, pointwise. By direct calculation for any x K n L n G n (x, y) = 0 L n e tl n δ x (y)dt = 0 t e tl n δ x (y)dt =[e tl n δ x (y)] 0 = δ x(y) and, hence, G n (x, ) areharmoniconk n \{x}, n 0. Let u be a non-trivial non-negative superharmonic function which is strictly positive by the minimum principle [35, Theorem 8].Let U X be finite with o U K n, n 0 and C > 0 be such that Cu G(o, ) on U. By the minimum principle Cu G n (o, ) on K n \{o} and, hence, Cu G(o, ) on X by the discussion above. If the graph is stochastically complete, then we get by Fubini s theorem C u 1 G(o, ) 1 = 0 e tl δ o (x)m(x)dt = x X 0 e tl 1(o)dt = 0 dt =. Hence, u is not in L 1 (X, m). In the proof we show that in the non-recurrent case there are no nontrivial superharmonic functions in L 1. This is explained since in the case of finite measure recurrence and stochastic completeness are equivalent [47, Theorem 12]. Next, we show that in general there is no L p Liouville theorem for p (0, 1]. This is analogous to the situation in Riemannian geometry, where counter-examples were given by [8,41]. Our first example is a graph of finite volume and the second is of infinite volume. Example 4.1 (Finite volume) Let G = (X,μ,m) be an infinite line graph, i.e., X = Z and x y iff x y =1forx, y Z. Define the edge weight by μ xy = 2 1 ( x y ) for x y and the measure m by m(x) = ( x +1) 2 2 x, x Z, which implies m(x) <. The intrinsic metric δ (introduced in Example 2.4) is compatible as it satisfies δ(x, x + 1) C( x +1) 1 and, thus, x= δ(x, x + 1) =. However, the function f defined as f (x) = sign(x)(2 x 1), x Z, is harmonic and, clearly, f L p (X, m), p (0, 1]. Example 4.2 (Infinite volume) We can extend the example above to the infinite volume case. Let G be the graph from above and G be a locally finite graph of infinite volume which allows for a compatible path metric. We glue G to the vertex x = 0 of the graph G by identifying a vertex in G with x = 0. Next, we extend the path metrics in the natural way and obtain (by renormalizing the edge weights of the metric at the edges around x = 0if necessary) again a compatible intrinsic metric and the graph has infinite volume. Moreover, we extend f on G from above by zero to G and obtain a harmonic function which is in L p, p (0, 1]. 123

209 356 B. Hua, M. Keller 5 Applications of Karp s theorem In this section we prove Theorem 1.10 and give several applications which mainly circle around the case of finite measure. Proof of Theorem 1.10 We assume that for some p (1, ) the non-negative subharmonic function f is in L p (X, mρ1 2 ) and, hence, f p ρ1 2 L 1 (X, m).forlarger 0 1, we estimate r 0 r f 1 Br p dr p r 0 r r 2 f p ρ1 2 1 dr C B r 1 r 0 1 r dr =. Hence, Theorem 1.1 implies that f is constant. Next, we turn to several consequences of Theorem A function f : X R is said to grow less than a function g :[0, ) (0, ) if there are β (0, 1) and C > 0 such that f (x) Cg β (ρ 1 (x)), x X. We say f grows polynomially if f grows less than a polynomial. We say the measure m has a finite q th moment, q R, with respect to an intrinsic metric ρ if ρ 1 L q (X, m), where ρ 1 = 1 ρ(, o). This assumption implies that all balls have finite measure and if q 0 it also implies m(x) <. Corollary 5.1 (Measures with finite moments) Assume a connected weighted graph allows for a compatible intrinsic metric and the measure has a finite qth moment, q R. Then every non-negative subharmonic function f that grows less than r r q+2 is constant. In particular, if q > 2, then boundedness of f implies f is constant. Proof If f growslessthanr r q+2, then there is ε>0such that f 1+ε ρ1 2 Cρ q 1 on X. By the assumption ρ 1 L q (X, m) it follows f L p (X, mρ1 2 ) for p = 1 + ε. Hence, the assertion follows from Theorem Letting q = 0 in the above theorem gives the following immediate corollary. Corollary 5.2 (Finite measure) Assume a connected weighted graph X allows for a compatible intrinsic metric and m(x) <. Then every non-negative subharmonic function f that grows less than quadratic is constant. In particular, f L (X) implies that f is constant. The final corollary of this section is a consequence of Corollary 1.2. Corollary 5.3 (Exponentially decaying measure) Assume a connected weighted graph X allows for a compatible intrinsic metric and m(x) <, and there is β>0such that 1 lim sup r r β log m(b r+1 \ B r )<0. Then every non-negative subharmonic function that grows polynomially is constant. 123

210 Harmonic functions of general graph Laplacians 357 Proof If a non-negative subharmonic function f grows polynomially, then there is q > 0 such that f p p C x X ρ q 1 (x, o)m(x) C r q m(b r \ B r 1 ) + C < r=1 by the assumption on the measure. Hence, the theorem follows from Corollary Applications to harmonic maps Harmonic maps between metric measure spaces were introduced by Jost [27 30] and harmonic maps from graphs into Riemannian manifolds or metric spaces have been studied by many authors, e.g. [26,31,39] and and for alternative definitions, see [19,37,38,50,52]. We use our results concerning the function theory on graphs to derive various Liouville type theorems for harmonic maps from graphs. A particular focus lies on bounded harmonic maps and harmonic maps of finite energy. Let (X,μ,m) be a weighted graph. We briefly recall the set up of Hadamard spaces and harmonic maps. A complete geodesic space (Y, d) is called an NPC space if it locally satisfies Toponogov s triangle comparison for non-positive sectional curvature. We refer to Burago Burago Ivanov [1], Jost [29] and Bridson Haefliger[2] for definitions. Here NPC stands for non-positive curvature in the sense of Alexandrov. The space (Y, d) is called an Hadamard space, if the Toponogov s triangle comparison holds globally, i.e., holds for arbitrary large geodesic triangles. A simply connected NPC space is an Hadamard space, see [1]. For the sake of simplicity, we only consider Hadamard spaces, also called CAT(0) spaces, as targets of harmonic maps X Y. For general NPC spaces, we may pass to the universal covers of X and Y, and consider the equivariant harmonic maps, see Jost [27,28]. Let b : P 1 (Y ) Y denote the barycenter map on Y,whereP 1 (Y ) is the space of probability measures on Y with finite first moment, that is, b(ν) is the barycenter of the probability measure ν on Y, see e.g. Sturm [51, Propositon 4.3]and confer [40, Definition 2.2 and Example 1]. We define the random walk measure P x of x X by P x (y) := μ xy z X μ xz and denote by u P x the push forward of the probability measure P x under the map u : X Y. In order to carry out the barycenter construction for a map u pointwise, we need u P x P 1 (Y ) which means that u P x has finite first moment. Thus, similar to the harmonic function case, we define a class of maps F(X, Y ) := u : X Y d(u(y), y 0 )P x (y) < for all x X, y 0 Y. y X Definition 6.1 (Harmonic map) A map u : X Y is called a harmonic map if u F(X, Y ) and for every x X u(x) = b(u P x ). 123

211 358 B. Hua, M. Keller One immediately finds that the measure m plays no role in the definition of harmonic maps. In the following, we always denote by u : X Y a harmonic map from a weighted graph into an Hadamard space. 6.1 Proof of Theorem 1.11 The proof of Theorem 1.11 is a rather immediate consequence of Theorem 1.10 and the following lemma which is a consequence of Jensen s inequality and convexity of distance functions on Hadamard spaces. Lemma 6.2 For every harmonic map u : X Y where Y is an Hadamard space, the functions X [0, ), x d(u(x), y), for fixed y Y, are subharmonic. Proof Jensen s inequality in Hadamard spaces, see [51, Theorem 6.2], states that for every lower semi-continuous convex function g : Y [0, ) and ν P 1 (Y ) g(b(ν)) g(y)ν(dy). Y Now any distance function y d(y, y 0 ) to a point y 0 Y is convex in an Hadamard space, see e.g. [1, Corollary ], which yields the statement. Proof of Theorem 1.11 Combining Theorem 1.10 and Lemma 6.2 yields that x d(u(x), y) is constant. Hence, u is bounded. If mρ1 2 (X) = x X m(x)ρ 2 1 (x) =, then a constant function is in L p (X, mρ1 2 ) if and only if it is zero. Moreover, if y is in the image of u then d(u( ), y) 0 and hence u(x) = y for all x X. 6.2 Harmonic maps of finite energy In this section we consider harmonic maps of finite energy and prove Theorems 1.12 and 1.13 which are analogues to theorems ofcheng Tam Wang [9] from Riemannian geometry. We say a harmonic map u : X Y has finite energy if 1 μ xy d 2 (u(x), u(y)) <. 2 x,y X In order to do so, we need the equivalence of boundedness of finite energy harmonic functions on a graph and that of non-negative subharmonic functions. Recall that a function f : X R is said to have finite energy if E( f )<, see Sect In Riemannian geometry such a theorem was first proven in [9, Theorem 1.2]. We give a different proof here in the discrete setting by Royden s decomposition. Theorem 6.3 For connected weighted graphs every harmonic function with finite energy is bounded if and only if every non-negative subharmonic function with finite energy is bounded. Proof As positive and negative part of a harmonic function are non-negative and subharmonic functions, boundedness of non-negative subharmonic functions of finite energy implies boundedness of harmonic functions of finite energy. We now turn to the other direction. By Proposition 3.3 there are no non-constant subharmonic functions of finite energy in the case the graph is recurrent. Therefore, we assume the graph is not recurrent (also called transient in the connected case). Let f be a non-negative subharmonic function with finite energy. Then by the discrete version of Royden s decomposition theorem, see [48, Theorem 3.69], 123

212 Harmonic functions of general graph Laplacians 359 there are unique functions g and h where g is in the completion of C c (X) under the norm ϕ o = (E(ϕ) + ϕ(o) 2 ) 1/2,ϕ C c (X), andh is a harmonic function of finite energy such that f = g + h and E( f ) = E(g) + E(h) By [48, Lemma 3.70], g 0sinceg is subharmonic. Therefore, 0 f h. By assumption h is bounded and, therefore, f is bounded. Proof of Theorem 1.12 Let u : X Y be a harmonic map of finite energy. For some fixed y 0 Y the function f = d(u( ), y 0 ) is non-negative and subharmonic by Lemma 6.2. Furthermore, by the triangle inequality and the assumption that u has finite energy we get E( f ) = 1 μ xy ( f (x) f (y)) 2 1 μ xy d 2 (u(x), u(y)) <. 2 2 x,y x,y X Now, by Theorem 6.3we get that f as a non-negative subharmonic function of finite energy must be boundedwheneverevery harmonic function of finite energy on X is bounded (which is our assumption). Thus, u is bounded. Theorem 1.13 is a consequence of Theorem 1.12 and the theorem of Kuwae Sturm [40] below which goes back to Kendall in the manifold case [33, Theorem 6] (confer [22,40,42]). However, although it is not explicitly mentioned in [40] one actually needs an additional assumption on the local compactness of the target, i.e., every point has a precompact neighborhood. Theorem 6.4 (Kendall s theorem [40, Theorem3.1]) Assume that on a connected weighted graph every bounded harmonic functions is constant. Then, every bounded harmonic map into a locally compact Hadamard space is constant. Next, we come to the proof of Theorem Proof of Theorem 1.13 Let f be a harmonic function on X of finite energy. By a discrete version of Virtanen s theorem, see [48, Theorem 3.73], f can be approximated by bounded harmonic functions f n of finite energy (with respect to the norm ϕ o = (E(ϕ)+ϕ(o) 2 ) 1/2 ). By assumption the functions f n, n 1, are constant and, thus, f must be constant. Theorem 1.12 implies now that any harmonic map is bounded and, thus, Theorem 6.4 implies that every harmonic map is constant. 6.3 Harmonic maps and assumptions on the measure of X In this subsection we collect several quantitative results that follow from what we have proven before. The first corollary can be seen as an analogue to Yau s L p -Liouville type theorem. Corollary 6.5 Assume a connected weighted graph X allows for a compatible intrinsic metric and let u be a harmonic map into an Hadamard space Y. If there is y Y such that d(u( ), y) L p (X, m) for some p (1, ), then u is bounded. If additionally m(x) =, then u is constant. Proof The function d(u( ), y) is subharmonic, by Lemma 6.2,andinL p (X, m) by assumption. Hence, Corollary 1.2 yields d(u( ), y) is constant. The assumption of infinite measure implies d(u(x), y) = 0forallx X. 123

213 360 B. Hua, M. Keller We say a harmonic map u into an Hadamard space (Y, d) grows less than a function g :[0, ) (0, ) if d(u( ), y) growslessthang for some y Y (confer Sect. 5). The next two corollaries are analogues of Corollaries 5.1 and 5.2. Corollary 6.6 (Measures with finite moment harmonic maps) Assume a connected weighted graph allows for a compatible intrinsic metric and the measure has a finite qth moment, q > 2. Then every harmonic map into an Hadamard space that grows less that r r q+2 is constant. In particular, bounded harmonic maps and harmonic maps with finite energy are constant. Proof Let u beaharmonicmap.sinceweassumeq + 2 > 0, we get by the triangle inequality that for all y Y the subharmonic (Lemma 6.2) functions d(u( ), y) grow less than r r q+2. Hence, by Corollary 5.1 the subharmonic function d(u( ), y) is constant for all y which implies that u is constant. This proves the first assertion. Since q + 2 > 0, it is easy to see that every bounded harmonic function on X is constant. The second assertion follows from Theorems 6.4 and Corollary 6.7 (Finite measure harmonic maps) Assume a connected weighted graph X allows for a compatible intrinsic metric and m(x) <. Then every harmonic map into an Hadamard space that grows less than quadratic is constant. In particular, bounded harmonic maps and harmonic maps with finite energy are constant. Proof The statements follow directly by the corollary above putting q = 0. Finally we say that a harmonic map grows polynomially if it grows less than a polynomial and state a corollary analogous to Corollary 5.3. Theorem 6.8 (Exponentially decaying measure harmonic maps) Assume a connected weighted graph X allows for a compatible intrinsic metric, m(x) < and there is β>0 such that lim sup r 1 r β log m(b r+1 \ B r )<0. Then every harmonic map into an Hadamard space that grows polynomially is constant. In particular, bounded harmonic maps and harmonic maps with finite energy are constant. Proof The statements follow from Corollary 5.3, Theorems 6.4 and Acknowledgments BH thanks Jürgen Jost for inspiring discussions on L p Liouville theorem and constant support, and acknowledges the financial support from the funding of the European Research Council under the European Union s Seventh Framework Programme (FP7/ ) / ERC grant agreement no MK enjoyed discussions with Gabor Lippner, Dan Mangoubi, Marcel Schmidt and Radosław Wojciechowski on the subject and acknowledges the financial support of the German Science Foundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (grant no. 1105/10 and no. 225/10) and BSF grant no References 1. Burago, D., Burago, Yu., Ivanov, S.: A Course in Metric Geometry. Number 33 in Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001) 2. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Number 319 in Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1999) 3. Bauer, F., Hua, B., Keller, M.: On the l p spectrum of Laplacians on graphs. Adv. Math. 248, (2013) 123

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217 CHAPTER 9 X. Huang, M. Keller, J. Masamune, R. Wojciechowski, A note on self-adjoint extensions of the Laplacian on weighted graphs, Journal of Functional Analysis 265 (2013),

218 Available online at Journal of Functional Analysis 265 (2013) A note on self-adjoint extensions of the Laplacian on weighted graphs Xueping Huang a, Matthias Keller b, Jun Masamune c, Radosław K. Wojciechowski d, a Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany b Mathematisches Institut, Friedrich-Schiller-Universität Jena, Jena, Germany c Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba, Aoba-ku, Sendai , Japan d Department of Mathematics and Computer Science, York College of the City University of New York, Guy R. Brewer Blvd., Jamaica, NY 11451, USA Received 31 August 2012; accepted 3 June 2013 Available online 5 July 2013 Communicated by Daniel W. Stroock Abstract We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions Elsevier Inc. All rights reserved. Keywords: Weighted graphs; Laplacians; Essential self-adjointness; Intrinsic metrics 1. Introduction Determining the uniqueness of self-adjoint extensions of a symmetric operator in a certain class is a fundamental topic of functional analysis going back to the work of Friedrichs and von Neumann [12,45]. If an operator has a unique self-adjoint extension, then it is called essentially * Corresponding author. addresses: xhuang1@math.uni-bielefeld.de (X. Huang), m.keller@uni-jena.de (M. Keller), jmasamune@m.tohoku.ac.jp (J. Masamune), rwojciechowski@gc.cuny.edu (R.K. Wojciechowski) /$ see front matter 2013 Elsevier Inc. All rights reserved.

219 X. Huang et al. / Journal of Functional Analysis 265 (2013) self-adjoint. A self-adjoint extension whose corresponding form is a Dirichlet form is called Markovian and, when such an extension is unique, the operator is said to have a unique Markovian extension. It is clear that essential self-adjointness implies the uniqueness of Markovian extensions, but the converse is not necessarily true as can be seen by examples. In the case of Riemannian manifolds, the (minimal) Laplacian, whose domain is the space of smooth functions with compact support, has Markovian extensions and generates the Brownian motion. (The Laplacian should satisfy, in addition, the regularity property, but there is always an equivalent operator which has this property [13].) A well-known result going back to the work of Gaffney [15,16] essentially states that, on a geodesically complete manifold, the Laplacian has a unique Markovian extension. (In [15,16] the so-called Gaffney Laplacian was proven to be essentially self-adjoint instead of the minimal one. The essential self-adjointness of the Gaffney Laplacian is equivalent to the uniqueness of Markovian extensions of the minimal Laplacian. Indeed, Gaffney s result states that the condition W 1,2 0 = W 1,2, which is equivalent to the uniqueness of Markovian extensions of the minimal Laplacian [22], implies the essential selfadjointness of the Gaffney Laplacian. The converse implication was proven in [38].) Later, it was shown that the Laplacian on a geodesically complete Riemannian manifold is essentially self-adjoint [2,43]. On the other hand, if the manifold is geodesically incomplete, the Laplacian is not essentially self-adjoint in general; however, if the Cauchy boundary, which is the difference between the completion of the manifold and the manifold itself, is small in some sense, then the Laplacian is essentially self-adjoint or has a unique Markov extension depending on how small the Cauchy boundary is [1,3,22,35 37] (see also the references therein). For strongly local regular Dirichlet forms, the uniqueness of Silverstein extensions was proven by Kawabata and Takeda [30] in the case when the underlying space is metrically complete with respect to the Carnot Caratheodori distance. This result was extended to general regular Dirichlet forms by Kuwae and Shiozawa [34] using the intrinsic distance defined by Frank, Lenz, and Wingert in [11]. Recently, there has been a tremendous amount of work devoted to the study of self-adjoint extensions of certain operators defined on graphs. More specifically, these issues are studied for adjacency, (magnetic) Laplacian, and Schrödinger-type operators on graphs in [4 6,17 19,24, 28,29,31,32,35,39,41,42,44,46 48] among others. Let us mention,in particular,the series of papers [4,5,44] by Colin de Verdière,Torki-Hamza, and Truc. These papers give some relations between metric completeness and essential selfadjointness. However, [24] contains an example of a graph which is metrically complete in one of the distances studied in [4] but for which the corresponding weighted Laplacian does not have a unique Markovian extension and is, therefore, not essentially self-adjoint. One reason for this seems to be that the particular metric used in [4] does not take into account the measure on the vertices of the graph. In [41,42], Milatovic, following [44], shows, with a different metric, that completeness implies essential self-adjointness under the additional assumption of a uniform bound on the vertex degree. In this paper we investigate these questions for the weighted Laplacian on graphs. Recall that the weighted Laplacian has Markovian extensions and the associated form is one of the most important examples of a non-local Dirichlet form. We use the notion of intrinsic distance introduced in [11] and show that, if a weighted degree function is bounded on the combinatorial neighborhood of each ball defined with respect to one such distance, then the Laplacian is essentially self-adjoint (Theorem 1). As a direct consequence, in the locally finite case, if the graph is metrically complete in one intrinsic path metric, then the Laplacian is essentially self-adjoint (Theorem 2). Compared to the previous results mentioned above we do not assume a uniform

220 1558 X. Huang et al. / Journal of Functional Analysis 265 (2013) bound on the vertex degree and, for Theorem 1, we do not even need local finiteness. These results indicate that intrinsic metrics give the correct notion of distance on graphs when seeking to prove statements analogous to the strongly local case. In the metrically incomplete case, under some further assumptions, we show that if the Cauchy (or metric) boundary has finite capacity, then the Laplacian has a unique Markovian extension if and only if the Cauchy boundary is polar, that is, has zero capacity, in analogy with [37] (Theorem 3). Moreover, we show that upper Minkowski codimension of the boundary greater than 2 implies zero capacity of the boundary (Theorem 4). We also show by examples that the other implications do not hold. In particular, in the case when the boundary has infinite capacity, the Laplacian may be essentially self-adjoint or might fail to have a unique Markovian extension, see Examples 5.2 and 5.4. In general, if the Laplacian is essentially self-adjoint, then it has a unique Markovian extension, but the opposite implication is not necessarily true, see Example 5.1. In Examples 5.5, 5.6, and 5.7 we discuss the case of upper Minkowski codimension less than or equal to 2 where the boundary may be polar or non-polar. The paper is organized as follows. In Section 2, we introduce the set up, including background material on Dirichlet forms, Laplacians, intrinsic distances, and Cauchy boundary; and state the main results. In Section 3, we establish the triviality of square integrable eigenfunctions with negative eigenvalue when the weighted degree function is bounded on the combinatorial neighborhood of each ball and use this to prove Theorems 1 and 2. Section 4 is devoted to the proofs of Theorems 3 and 4 and Section 5 is devoted to (counter-)examples. In Appendix A, we prove a Hopf Rinow type property for path metrics on locally finite graphs. This property is used in the proof of Theorems 2 and 3. We also present a series of (counter-)examples showing that the property may fail if the graph is not locally finite. 2. The set up and main results 2.1. Weighted graphs We generally follow the setting of [31]. LetX be a countably infinite discrete set. Elements of X will be called vertices. A function μ : X (0, ) can be viewed as a Radon measure on X with full support so that (X, μ) becomes a measure space. Let w : X X [0, ) be symmetric, with zero diagonal, and satisfying w(x,y) < for all x X. y X The triple (X,w,μ) is called a weighted graph. We call x,y X neighbors if w(x,y) > 0 and denote this symmetric relation by x y. If each vertex has only finitely many neighbors, then the graph is called locally finite. Forn 1, we call a sequence of points (x 0,x 1,...,x n ) a path connecting x and y if x 0 = x, x n = y, and x i x i+1 for i = 0, 1,...,n 1. A weighted graph (X,w,μ) is called connected if, for any two distinct points in X, there exists a connecting path. From now on, we only consider connected weighted graphs Weighted degree and intrinsic metrics We call the function Deg : X [0, ) given by Deg(x) := 1 μ(x) w(x,y) y X

221 X. Huang et al. / Journal of Functional Analysis 265 (2013) the weighted degree. It is, in general, distinct from the combinatorial degree of locally finite graphs which is given by the number of neighbors of a vertex. A pseudo metric is a map d : X X [0, ) that is symmetric, has zero diagonal and satisfies the triangle inequality. A pseudo metric d = d σ is called a path pseudo metric if there is a symmetric map σ : X X [0, ) such that σ(x,y) > 0 if and only if x y and d σ (x, y) = inf { l σ ( (x0,...,x n ) ) n 1, (x0,...,x n ) is a path connecting x and y } where the length l σ of a path (x 0,...,x n ) is given by n 1 ( l σ (x0,...,x n ) ) = σ(x i,x i+1 ). We say that a pseudo metric d has jump size s>0 if, for all x,y X, w(x,y) = 0 whenever d(x,y) > s. Following Frank, Lenz and Wingert [11] (see Lemma 4.7 and Theorem 7.3) we make a definition which has already proven to be useful in several other problems on graphs, see Remark 2.2 below. Definition. We call a pseudo metric d on (X,w,μ)intrinsic if, for all x X, 1 μ(x) i=0 w(x,y)d(x,y) 2 1. y X An intrinsic path pseudo metric d σ is called strongly intrinsic if, for all x X, 1 μ(x) w(x,y)σ(x,y) 2 1. y X The first example below shows that there always exist strongly intrinsic path pseudo metrics with jump size 1 on a connected weighted graph. Example 2.1. (1) For x,y X with x y,letσ 0 (x, y) = min{deg 1 2 (x), Deg 1 2 (y), 1}. Clearly, d σ0 is strongly intrinsic with jump size 1. (2) For locally finite graphs, let σ 1 (x, y) = w(x,y) 1 2 min{ μ(x) deg(x), μ(y) deg(y) } 1 2, x,y X with x y where deg is the combinatorial degree, i.e., the number of neighbors. Clearly, d σ1 is a strongly intrinsic path metric. Moreover, if deg K for some K 1, then d σ1 is equivalent to the metrics used in [4,5,41,42] (in the case of no magnetic field and no potential). This seems to explain why the combinatorial vertex degree has to be bounded for these results. (3) Suppose that σ N 1 on neighbors. Then, d N = d σn gives the natural graph metric, that is, the distance between x and y is equal to one less than the number of points in the shortest path connecting them. Obviously, d N is strongly intrinsic if and only if Deg 1. (Clearly, if Deg is bounded by K>0, then d N / K is also a strongly intrinsic metric.)

222 1560 X. Huang et al. / Journal of Functional Analysis 265 (2013) Remark 2.2. Various authors came up with such types of metrics independently of [11]. In the context of stochastic completeness for jump processes, see the work of Masamune and Uemura [40], Grigor yan, Huang and Masamune [21] and also [26,27]. Independently, Folz [8] came up with similar ideas in the context of heat kernel estimates on locally finite graphs, see also [9,10]. For further uses of intrinsic metrics, see [25]. For x 0 X and r 0, we define the distance ball with respect to any pseudo metric d by B r (x 0 ) := {x X d(x,x 0 ) r} Forms and operators In this article, we only consider real valued functions. Denote by C(X) the set of all functions X R and by C c (X) the subset of functions which are finitely supported. The Hilbert space L 2 (X, μ) is defined in the usual way with scalar product u, v := X uvμ := x X u(x)v(x)μ(x) and norm u := u, u 1 2 = ( X u2 μ) 1 2. We next introduce a discrete version of the energy measure which can be thought of as a generalized gradient. For f C(X) and x X define the square of the generalized gradient by f 2 (x) := y X w(x,y) ( f(x) f(y) ) 2, which might take the value. For x X, let D loc (x) := {(f, g) C(X) C(X) y X w(x,y) f(x) f(y) g(x) g(y) < } and for (f, g) D loc(x), we define ( f g)(x) := y X w(x,y) ( f(x) f(y) )( g(x) g(y) ). The generalized form Q is a map C(X) [0, ] given by Q(f ) := 1 f 2 = X x,y X w(x,y) ( f(x) f(y) ) 2 and the generalized form domain is given by D := {f C(X) Q(f ) < }. Clearly, C c (X) D. By polarization, this gives a sesqui-linear form Q : D D R as follows Q(f, g) = 1 ( f g) = X x,y X w(x,y) ( f(x) f(y) )( g(x) g(y) ). In this context, there are two distinguished restrictions of the generalized form. Let Q be the restriction of Q to D(Q) := C c (X) Q where Q := ( Q( ) + 2) 1 2.

223 X. Huang et al. / Journal of Functional Analysis 265 (2013) The form (Q, D(Q)) is then a regular Dirichlet form, see [31]. Furthermore, let Q max be the restriction of Q to D ( Q max) := { f L 2 (X, μ) Q(f ) < }. The form (Q max,d(q max )) is a Dirichlet form but it is not regular in general. For more discussion of these two forms and the associated self-adjoint operators in our context, see [24]. The formal Laplacian can be introduced on (X,w,μ) as an analogue of the classical Laplace Beltrami operator on Riemannian manifolds as follows ( f )(x) = 1 μ(x) w(x,y) ( f(x) f(y) ), y X with domain F ={f C(X) y X w(x,y) f(y) < for all x X}. Taking into account y w(x,y) <, x X, the operator is defined pointwise. It is easy to see that F is stable under multiplication by bounded functions on X. It can be shown that the self-adjoint operator L with domain D(L) corresponding to Q is non-negative and is a restriction of, see[31, Theorem 9]. That is, Lu = u, u D(L) Main results As discussed in the introduction, it is a classical result that the Laplacian on a weighted manifold is essentially self-adjoint if all geodesic balls are relatively compact which is equivalent to the manifold being metrically complete (see, for example, Theorem 11.5 in [20]). Here we present some counterparts for weighted graphs. We define the combinatorial neighborhood n(k) of a subset K of X by n(k) ={x X x K or there exists y K such that x y}. Note that the combinatorial neighborhood can be understood as the distance one ball about K with respect to the natural graph distance. Theorem 1. Let (X,w,μ) be a weighted graph and let d be an intrinsic pseudo metric. If the weighted degree function Deg is bounded on the combinatorial neighborhood of each distance ball, then D(Q) = D ( Q max), D(L)= { u L 2 (X, μ) F u L 2 (X, μ) }. In particular, if additionally C c (X) L 2 (X, μ), then L c = Cc (X) is essentially self-adjoint. Remark 2.3. (a) The result on essential self-adjointness is sharp by Example 5.1 in Section 5. (b) Let us note that the theorem does not assume that Deg is bounded on X. This would imply that Q is bounded and the statements become trivial. (c) The condition C c (X) L 2 (X, μ) holds if and only if w(x, )/μ( ) L 2 (X, μ) for all x X, see Proposition 3.3 in [31]. In particular, this always holds in the locally finite case.

224 1562 X. Huang et al. / Journal of Functional Analysis 265 (2013) If a pseudo metric has finite jump size, the combinatorial neighborhood of a distance ball is contained in another distance ball. This yields the following immediate consequence. Corollary 1. If (X,w,μ) is a weighted graph and d an intrinsic pseudo metric with finite jump size such that each distance ball is finite, then the conclusions of Theorem 1 hold. Note that finite balls and finite jump size imply that the graph is locally finite. In this case, path pseudo metrics are metrics and the analogy to Riemannian manifolds becomes even more obvious as can be seen below. Recall that an extension of L c = Cc (X) is said to be Markovian, if the form associated to it is a Dirichlet form. In particular, the operators associated to Q and Q max are Markovian and, in the locally finite case, having a unique Markovian extension is equivalent to D(Q) = D(Q max ) [24, Theorem 5.2]. Theorem 2. If (X,w,μ) is a locally finite weighted graph and d = d σ an intrinsic path metric such that (X, d) is metrically complete, then D(L) = { u L 2 (X, μ) u L 2 (X, μ) }, L c = Cc (X) is essentially self-adjoint and has a unique Markovian extension. Next, we turn to the metrically incomplete case where we will prove an analogue to results found in [37], seetheorem 3 below. In order to avoid some topological issues when defining capacity, we now assume that all graphs are locally finite and that we only deal with path metrics. Note that by (a) of Lemma A.3, the topology induced by a path metric is discrete in the locally finite case. For a set U X define the capacity of U by Cap(U) := inf { u Q u D ( Q max ), 1 U u }, where 1 U is the characteristic function of U and inf =. For a path metric d on X we let (X,d) be the metric completion. We define the Cauchy boundary C X of X to be the difference between X and X: C X := X \ X. Clearly, X is metrically complete if and only if C X is empty. For A X define Cap(A) := inf { Cap(O X) A O with O X open }. Note that, for U X, the definitions of capacity agree due to the local finiteness and the use of path metrics. We say that C X is polar if Cap( C X) = 0. Theorem 3. Let (X,w,μ)be a locally finite weighted graph and let d = d σ be a strongly intrinsic path metric. If (X, d) is not metrically complete and Cap( C X) <, then the Cauchy boundary is polar if and only if

225 X. Huang et al. / Journal of Functional Analysis 265 (2013) D(Q) = D ( Q max), that is, if and only if L c = Cc (X) has a unique Markovian extension. Note that the other consequences of Theorem 1 do not necessarily follow if (X, d) is metrically incomplete with polar boundary. Example 5.1 in Section 5 contains a weighted graph with polar boundary but where L c has non-markovian self-adjoint extensions. Also, in the case when the Cauchy boundary has infinite capacity, the Laplacian may be essentially self-adjoint or may not have a unique Markovian extension, see Examples 5.2 and 5.4. Next we turn to a criterion which connects polarity of the boundary to codimension of the boundary. The upper Minkowski codimension codim M ( C X) of C X is defined as codim M ( C X) = lim sup r 0 ln μ(b r ( C X)), ln r where B r ( C X) ={x X inf b C X d(x,b) r}. The upper Minkowski codimension (or box counting codimension) is one of the most studied fractal dimensions. The relationships between various dimensions in the classical setting can be found in [7]. Theorem 4. Let (X,w,μ)be a locally finite weighted graph and let d = d σ be an intrinsic path metric. If codim M ( C X) > 2, then C X is polar. In Examples 5.5, 5.6, and 5.7 we show that C X can be polar or non-polar if codim M ( C X) 2. See the recent paper [22] for some related statements in the case of manifolds. 3. Uniqueness of solutions Let (X,w,μ) be a weighted graph and let d be an intrinsic pseudo metric. In this section we will show that under the assumption that Deg is bounded on the combinatorial neighborhood of each distance ball, there are no non-trivial L 2 solutions to ( + λ)u = 0forλ>0. From this fact we can infer Theorems 1 and Leibniz rule and Green s formula The following auxiliary lemmas are well known in various other situations, see [11,23,24,28, 29,31,32]. However, they do not hold on graphs without further assumptions. Here, we prove them under the assumption that the weighted vertex degree is bounded on certain subsets. Lemma 3.1. Let B X be such that Deg is bounded on B. Then, for all u, v L 2 (X, μ), x,y B w(x,y) u(x)v(x) < and x,y B w(x,y) u(x)v(y) <. Proof. Let f = max{ u, v }. Clearly, f L 2 (X, μ). We estimate

226 1564 X. Huang et al. / Journal of Functional Analysis 265 (2013) x,y B w(x,y) u(x) v(x) x B f 2 (x) y B w(x,y) x B sup Deg(y) f 2 <, y B f 2 (x)μ(x) Deg(x) since Deg is bounded on B by assumption. On the other hand, by the Cauchy Schwarz inequality and the above, we obtain x,y B w(x,y) u(x) v(y) ( x,y B ) 1 ( w(x,y)u 2 2 (x) x,y B The following is an integrated Leibniz rule (see also [11, Theorem 3.7]). w(x,y)v 2 (x)) 1 2 <. Lemma 3.2 (Leibniz rule). Let U X be such that Deg is bounded on n(u). For all f L (X) with supp f U and g,h L 2 (X, μ) ( ) (fg) h = f( g h) + g( f h). X X X Proof. By means of Lemma 3.1 and the fact that supp f U, it is not hard to see that (fg, h), (f, h) D loc (x) for x n(u), and that (g, h) D loc (x) for x U. Moreover, all of the sums over X above are, in fact, sums over n(u) and the first sum on the right-hand side is over U. Hence, by basic estimates, Lemma 3.1 and the fact supp f U, all of the sums above converge absolutely. Therefore, the statement follows by the simple algebraic manipulation fg(x) fg(y)= f (x)(g(x) g(y)) + g(y)(f (x) f(y)), x,y X. The following Green s formula is a variant of [23, Lemma 4.7]. Lemma 3.3 (Green s formula). Assume that Deg is bounded on n(u) for some set U X. Then, for all u, v L 2 (X, μ) F with supp v U, ( u)vμ = u( v)μ = 1 ( u v). 2 X X X Proof. Since w(x,y) = 0 whenever x U,y X\n(U), the statement follows by simple algebraic manipulations, Lemma 3.1 and Fubini s theorem A Caccioppoli-type inequality The key estimate for the proof of triviality of L 2 solutions to ( + λ)u = 0forλ>0isthe following Caccioppoli-type inequality. See [11, Theorem 11.1] for a similar result for general Dirichlet forms. Lemma 3.4 (Caccioppoli-type inequality). Let u L 2 (X, μ) F, U X and assume that Deg is bounded on n(u). Then, for all v L (X) with supp v U, X ( u)uv 2 μ 1 u 2 v 2. 2 X

227 X. Huang et al. / Journal of Functional Analysis 265 (2013) Proof. By Lemmas 3.2 and 3.3 we have ( u)uv 2 μ = 1 ( ( u uv 2 )) = 1 v 2 u u ( u v 2) X X We focus on the second sum on the right-hand side. Since a geometric mean can be estimated by its corresponding arithmetic mean, we have ab δ 2 a δ b2 for a,b R and δ>0. We use this estimate with a = (u(x) u(y))(v(x) + v(y)), b = u(x)(v(x) v(y)) and δ = 1/2 forthe terms in the second sum on the right-hand side above u(x) ( u(x) u(y) )( v 2 (x) v 2 (y) ) 1 ( ) 2 ( ) 2 u(x) u(y) v(x) + v(y) + u 2 (x) ( v(x) v(y) ) ( u(x) u(y) ) 2 ( v 2 (x) + v 2 (y) ) + u 2 (x) ( v(x) v(y) ) 2. Multiplying by w(x,y) and summing over x,y X yields 1 u ( u v 2) 1 v 2 u u 2 v X X X X X The assertion now follows from the equality in the beginning of the proof Uniqueness of solutions The following is an analogue of [20, Lemma 11.6]. Proposition 3.5. Assume that the weighted degree function Deg is bounded on the combinatorial neighborhood of each distance ball. Then, for all λ>0, the equation has only the trivial solution in L 2 (X, μ) F. ( + λ)u = 0, Proof. Let 0 r<r,fixx 0 X, and consider the cut-off function η = η R,r : X R given by ( ) R d(x,x0 ) η(x) = R r Note that 0 η 1, η Br = 1 and η X\BR = 0. Moreover, η is Lipshitz continuous with Lipshitz 1 constant R r (of course, with respect to the intrinsic pseudo metric d). This immediately implies, as d is an intrinsic pseudo metric, that + 1. η 2 (x) 1 (R r) 2 w(x,y)d 2 (x, y) y X μ(x) (R r) 2, x X.

228 1566 X. Huang et al. / Journal of Functional Analysis 265 (2013) Fix λ>0 and assume that u L 2 (X, μ) F is a solution to the equation ( + λ)u = 0. Then, we have by Lemma 3.4 and the estimate on η 2 above, that λ u1 Br 2 λ uη 2 = X ( u)uη 2 μ 1 u 2 η 2 2 X 1 2(R r) 2 u 2. Letting R, we see that u 0onB r. Since r is chosen arbitrarily, u 0onX. Remark 3.6. It would be interesting to prove a similar statement as Proposition 3.5 for L p Proofs of Theorems 1 and 2 The proof of Theorem 1 follows by standard techniques used in [31] and [24]. Proof of Theorem 1. By [24, Corollary 4.3], D(Q) = D(Q max ) is equivalent to the nonexistence of non-trivial solutions to ( + λ)u = 0forλ>0inD(Q max ). Thus, D(Q) = D(Q max ) follows from Proposition 3.5. Define D max ={u L 2 (X, μ) F u L 2 (X, μ)}. By Theorem 9 in [31], L is a restriction of which implies that D(L) D max. Letting f D max we see that g := ( + λ)f L 2 (X, μ) for all λ>0, so that u := (L + λ) 1 g D(L). Asu solves the equation ( + λ)u = g (see Lemma 2.8 in [31]), we conclude that f = u by the uniqueness of solutions, Proposition 3.5 (as f solves the equation by definition). Thus, f D(L) and, therefore, D(L) = D max. Assuming C c (X) L 2 (X, μ), essential self-adjointness is a rather immediate consequence of L = L c. By Green s formula for functions in v C c(x) and f F (see [23, Lemma 4.7] or [31, Proposition 3.3]), we have X f(l cv)μ = X ( f )vμ and thus D(L c ) = D max. Hence, by what we have shown above, we have D(L c ) = D(L) and, therefore, it follows that L = L c. Proof of Theorem 2. As we assume local finiteness, it is clear that C c (X) L 2 (X, μ). Furthermore, by Theorem A.1, the metric completeness of (X, d) implies that distance balls are finite. Note that the combinatorial neighborhood of a finite set is again finite. Hence, Deg is bounded on the combinatorial neighborhood of each distance ball which implies the statements about essential self-adjointness and D(Q) = D(Q max ) by Theorem 1. As uniqueness of Markovian extensions is equivalent to D(Q) = D(Q max ) in the locally finite case, see [24, Theorem 4.2], the second statement follows as well. 4. Cauchy boundary and equilibrium potentials Let (X,w,μ)be a locally finite weighted graph and let d be a path metric. Recall that (X,d) denotes the metric completion of (X, d) and C X = X \ X denotes the Cauchy boundary. In this section we prove Theorems 3 and Existence of equilibrium potentials The following is well known and follows directly from [14, Lemma 2.1.1]. Lemma 4.1. If Cap(O) < for an open set O X, then there is a unique element e D(Q max ) such that 0 e 1, e O X 1, and Cap(O) = e Q.

229 X. Huang et al. / Journal of Functional Analysis 265 (2013) Proof. From [14, Lemma 2.1.1] it follows that for any U X there is such an e D(Q max ) (as we consider X equipped with the discrete topology). Note that in [14, Lemma 2.1.1] regularity of the form is a standing assumption but this is not needed for the proof. Now, for an open set O X the equality Cap(O) = Cap(O X) follows from the definition (by taking A = O). Hence, we let e for O be the corresponding e for O X. We call such an e the equilibrium potential associated to O The boundary alternative The following lemma shows that if the minimal and maximal forms agree, then the capacity of any subset of the boundary is either zero or infinite. Lemma 4.2. Let A C X.IfD(Q) = D(Q max ), then either Cap(A) = or Cap(A) = 0. Proof. Assume that D(Q) = D(Q max ) and A has finite capacity. Then, there exists an open set O X such that A O and Cap(O) <. Lete be the equilibrium potential associated to O. Since D(Q) = D(Q max ) there exists a sequence of functions e n in C c (X) converging to e as n in the Q norm. Clearly, (e e n) + 1 belongs to D(Q max ) and equals 1 on O n X, where O n is a neighborhood of A in X. Therefore, Cap(A) lim inf n Cap(O n) lim n (e e n ) + 1 Q lim n e e n Q = Approximation by equilibrium potentials Next, we show that every bounded function in D(Q max ) can be approximated via equilibrium potentials if the boundary has capacity zero. Lemma 4.3. Assume that C X is polar and let e n be the equilibrium potentials associated to open sets O n X with C X O n and Cap(O n ) 0 as n. Then u (1 e n )u Q 0 as n for all u D(Q max ) L (X). Proof. Note that u (1 e n )u = e n u and e n u u e n 0asn. Moreover, we have Q(e n u) = 1 (en u) 2 2 X X e 2 n u 2 + X u 2 e n 2 X e 2 n u u 2 Q(e n ) 0 asn by noting that e n (x) 0 for all x and applying the Lebesgue dominated convergence theorem.

230 1568 X. Huang et al. / Journal of Functional Analysis 265 (2013) Restriction to complete subgraphs In the next lemma we show that bounded functions in D(Q max ) that are zero close to the boundary can be approximated by finitely supported functions. We show this by restricting our attention to complete subgraphs. In order for the restriction of an intrinsic metric to be intrinsic, we need to assume that the metric is strongly intrinsic. Lemma 4.4. Assume that d = d σ is a strongly intrinsic path metric. Let O X be open with C X O, Cap(O) < and let e be the equilibrium potential associated to O. Then, C c (X) is dense in (1 e)(d(q max ) L (X)) ={(1 e)u u D(Q max ) L (X)} with respect to Q. Proof. Let Y = X \O, μ Y be the restriction of μ to Y and σ Y and w Y be the restrictions of σ and w to Y Y. We first assume that (Y, w Y,μ Y ) is connected. From this it follows that d Y = d σy is a strongly intrinsic path metric on (Y, w Y,μ Y ) with d Y d. Claim. (Y, d Y ) is metrically complete. Proof. If (x n ) is a Cauchy sequence in Y, then, by d Y d, it is a Cauchy sequence in X and has a limit point in X. However, as Y = X \ O, the limit point is not in C X.As(X, d σ ) is a discrete metric space, see Lemma A.3(a), (x n ) must be eventually constant which proves the claim. For R>0 and fixed x 0 Y,letη R : Y [0, 1] given by ( ) 2R dy (x, x 0 ) η R (x) = R By completeness, B 2R in (Y, d Y ) is finite, see Theorem A.1 in Appendix A, and it follows that η R (1 e)d(q max ) C c (X). Let be the generalized gradient for (Y, w Y,μ Y ).Letv (1 e)(d(q max ) L (X)) and set g R = v η R v = (1 η R )v.now, Q(g R ) = 1 g R Y x Y y O X + 1. w(x,y)g 2 R (x). For the first term we get, using that d Y is intrinsic and, therefore, that η R 2 μ Y /R 2, 1 2 g R 2 Y Y v 2 η R 2 + Y (1 η R ) 2 v 2 1 R 2 v 2 + Y (1 η R ) 2 v 2 0, as R. For the second term let u (D(Q max ) L (X)) such that v = (1 e)u. Then, w(x,y)g R (x) 2 u 2 ( 1 ηr (x) ) 2 w(x,y) ( 1 e(x) ) 2 0 x Y y O X x Y y O X as R by the Lebesgue dominated convergence theorem. This follows, since η R 1 pointwise as R and Y O X w(x,y)(1 e(x))2 Q(e) Cap(O) 2 < (as e(y) = 1for y O X). Moreover, as g R converges pointwise to zero it also converges to zero in L 2.

231 X. Huang et al. / Journal of Functional Analysis 265 (2013) In the case where Y is not connected there are at most countably many connected components Y i, i 0. For v (1 e)(d(q max ) L (X)) let v i = v Yi. Since Q(v) = i 0 Q(v i ), the statement follows by a diagonal sequence argument Proof of Theorem 3 Proof of Theorem 3. Since we assume local finiteness, L c having a unique Markovian extension is equivalent to D(Q) = D(Q max ) by Theorem 5.2 in [24]. If D(Q) = D(Q max ), then the assumption Cap( C X) < implies Cap( C X) = 0 by Lemma 4.2. If, on the other hand, C X has zero capacity, then C c (X) is dense in D(Q max ) L (X) with respect to Q by Lemmas 4.3 and 4.4.ButD(Qmax ) L (X) is dense in D(Q max ) with respect to Q since if u D(Qmax ), then u n = (u n) n converges to u in Q as n (cf. [14, Theorem 1.4.2(iii)]). This implies D(Q) = D(Q max ) Proof of Theorem 4 Proof of Theorem 4. As codim M ( C X) > 2, there exists an ε>0 and a sequence r n 0as n such that μ ( B rn ( C X) ) <r 2+ε n. For R>0 and x X, let ( ) 2R d(x, C X) η R (x) = R + 1. In particular, 0 η R 1, η R BR ( C X) 1 and η R X\B2R ( C X) 0. It follows that and Q(η R ) x B 2R ( C X) y X 1 R 2 η R 2 μ ( B 2R ( C X) ) x B 2R ( C X) y X ( d(x, C X) w(x,y) d(y, ) 2 CX) R R since d is intrinsic. Applying the above with r n /2 in place of R, it follows that w(x,y)d(x,y) 2 μ(b 2R( C X)) R 2 ( Cap( C X) η rn /2 Q μ ( B rn ( C X) ) + 4 rn 2 μ ( B rn ( C X) )) 1 2 ( r 2+ε n + 4r ε n) asn.

232 1570 X. Huang et al. / Journal of Functional Analysis 265 (2013) (Counter-)examples Here we present the examples mentioned in Section 2.4. In particular, we show that Markov uniqueness does not imply essential self-adjointness, that no conclusion can be drawn concerning uniqueness in the infinite capacity case and that the boundary can be polar or non-polar for any upper Minkowski codimension less than or equal to 2. As we often use a graph with X = N 0 and x y if and only if x y =1 we make several preliminary observations concerning graphs of this type with a given path metric d. First,inthis case, C X if and only if l(x) := d(x,x + 1)<, x=0 see Theorem A.1 in Appendix A. Second, if C X, then Cap( C X) < if and only if μ(x) <. This can be seen as follows: if μ(x) =, then every neighborhood of the boundary must have infinite measure so that Cap( C X) =.Ifμ(X) <, then 1 D(Q max ) which implies that Cap( C X) 1 Q = μ(x) <. These two observations will be used repeatedly below. Example 5.1. (Polar Cauchy boundary (and consequently D(Q) = D(Q max )) but no essential selfadjointness.) Let X = Z with w(x,y) = 1if x y =1 and 0 otherwise. The strongly intrinsic path metric d = d σ0 introduced in Example 2.1 satisfies d(x,x + 1) = min{ μ(x)/2, μ(x + 1)/2, 1} for an arbitrary measure μ. Therefore, if the measure is chosen so that it satisfies x= x 2 μ(x) <, then (X, d) is metrically incomplete, the Cauchy boundary consists of two points and h : x x is in L 2 (X, μ) which we will use later. Define e n by One checks that e n D(Q max ) with Q(e n ) = e n (x) := ( x /n 1 ) + 1. x= ( en (x) e n (x + 1) ) 2 1 = 2n n 2 0 and that e n 0inL 2 (X, μ) as n by the Lebesgue dominated convergence theorem. Thus, the Cauchy boundary of X is polar and L c = Cc (X) has a unique Markovian extension. On the other hand, the formal Laplacian acts as f (x) = μ(x) 1 (f (x) f(x 1) + f(x) f(x+1)). Clearly, h(x) = x is harmonic, square integrable by the choice of μ, and h/ D(Q max ). This shows that h D(L c ) \ D(Qmax ), that is, L c is not essentially self-adjoint. Example 5.2. (Cauchy boundary with infinite capacity and essential self-adjointness.) Let X = N 0 with μ(x) = and w symmetric such that w(x,y) > 0 if and only if x y =1. By [31, Theorem 6] the operator Cc (X) is essentially self-adjoint. (This can be also seen directly as there are no non-trivial solutions to ( + λ)u = 0inL 2 (X, μ) for λ>0. This follows as any

233 X. Huang et al. / Journal of Functional Analysis 265 (2013) positive solution to this equation must be increasing by a minimum principle, see also Eq. (1) below.) If d = d σ0 and w and μ are chosen to satisfy l(x) = lim d(0,x) ( ) ( ) 1 μ(x) 2 <, x Deg(x) w(x,x + 1) x=0 then it follows that (X, d) is not metrically complete and that the boundary consists of a single point. Since μ(x) =,Cap( C X) = as noted above. Example 5.3. (Cauchy boundary with finite positive capacity and consequently D(Q) D(Q max ).) Let X = N 0 with μ(x) < and let w be symmetric with w(x,y) > 0 if and only if x y =1 and satisfying l(x) = lim x d(0,x) x=0 ( μ(x) w(x,x + 1) x=0 ) 1 2 < and x=0 1 w(x,x + 1) < where d = d σ0. In particular, the Cauchy boundary C X of X consists of one point and has finite capacity. Recall that D(Q) D(Q max ) is equivalent to ( + λ)u = 0 having a non-trivial solution in D(Q max ) for any λ>0 [24, Corollary 4.3]. By[33, Lemma 4.3], the equation ( + λ)u = 0on X translates to u(x + 1) u(x) = λ w(x,x + 1) from which it follows, see [33, Lemma 5.4], that u is bounded if and only if As μ(x) <, this is equivalent to x=0 x=0 x u(y)μ(y) (1) y=0 xy=0 μ(y) w(x,x + 1) <. 1 w(x,x + 1) <. Furthermore, as μ(x) <, u L (X) implies that u L 2 (X, μ). It is also not difficult to see that Q(u) < in this case as, by (1), we get that w(x,x + 1) ( u(x + 1) u(x) ) 2 1 ( ) 2. λμ(x) u w(x,x + 1) Therefore, as u D(Q max ) is non-trivial, D(Q) D(Q max ). Finally, Cap( C X) > 0 follows by combining Cap( C X) <, D(Q) D(Q max ), and Theorem 3.

234 1572 X. Huang et al. / Journal of Functional Analysis 265 (2013) Example 5.4. (Cauchy boundary with infinite capacity and D(Q) D(Q max ) (and consequently no essential self-adjointness).) We consider X = Z with X = X X + where X = N 0 with w and μ chosen as in Example 5.2 and X + = N 0 with w and μ chosen as in Example 5.3. In particular, the Cauchy boundary C X of X consists of two points, p L and p R, and has infinite capacity as Cap(p L ) = by μ(x ) =. On the other hand, by Example 5.3 we have 0 < Cap(p R )< which gives D(Q) D(Q max ) by Lemma 4.2. Example 5.5. (Polar Cauchy boundary with upper Minkowski codimension 2.) Let X = N 0 with w(x,y) = 1/8 if x y =1 and 0 otherwise and μ(x) = 4 x. Therefore, for x>0, Deg(x) = 4 x 1 so that, with d = d σ0, we get d(x,x + 1) = 2 x. Furthermore, by using the technique of Example 5.1, we can show that the Cauchy boundary consists of a single point, p R, and that Cap(p R ) = 0. Let r(x) := d(x,p R ) = y=x 2 y = 2 (x 1) so that μ(b r(x) (p R )) = y=x 4 y = 4 (x 1) /3 = r(x) 2 /3. Therefore, ln μ(b r(x) (p R )) ln r(x) = 2lnr(x) ln 3 ln r(x) 2 asx. Example 5.6. (Non-polar Cauchy boundary with upper Minkowski codimension 2.) Let X = N 0, with w symmetric, satisfying w(x,x + 1) = (x + 1) 2 and 0 otherwise with d(x,x + 1) = 1 (x + 1)2 2 x+2 and μ(x) = 4 x. It is easy to check that this metric is intrinsic. As l(x) <, μ(x) <, and x=0 1 w(x,x+1) < it follows that 0 < Cap( C X) < by the reasoning of Example 5.3. Therefore, codim M ( C X) 2byTheorem 4. By definition, r(x) := d(x, C X) = 1 2 x+1 and μ ( B r(x) ( C X) ) = y=x (y + 1) 2 4 y. Now, for every β>1/4, there exists an M such that (x + 1) 2 (4β) x for all x M. Hence, for all x M and 1/4 <β<1, we have μ(b r(x) ( C X)) y=x β y = β x (1 β) 1. Therefore, for all x M, ln μ(b r(x) ( C X)) ln r(x) ln(βx (1 β) 1 ) ln 2 (x+1) which implies that codim M ( C X) ln β/ln 2. As β>1/4 was chosen arbitrarily, it follows that codim M ( C X) 2 yielding that codim M ( C X) = 2. Example 5.7. (Upper Minkowski codimension between 0 and 2 with polar and non-polar Cauchy boundary.) Let X = N 0 with w(x,y) > 0 if and only if x y =1. Let, for α R,

235 X. Huang et al. / Journal of Functional Analysis 265 (2013) d(x,x + 1) = 1 2 α(x+1) and μ(x) = 1 2 (2α 1)x. Then, l(x) < for α>0 and μ(x) < for α>1/2. Thus, Cap( C X) < for α>1/2. Now, and so that r(x) := d(x, C X) = μ ( B r(x) ( C X) ) = y=x y=x ( ) α(y+1) = 2 α 1 2 αx ( 1 2 (2α 1)y = 1 2 2α 1 1 ) 1 2 (2α 1)(x 1) codim M ( C X) = 2 1 α. We now specify two choices of weights w: Case 1 polar Cauchy boundary: Letw(x,x + 1) = 1 for all x N 0. Clearly, d is intrinsic for all α>0. Furthermore, if Cap( C X) <, then Cap( C X) = 0asinExample 5.1. Hence, there exist examples of graphs with polar Cauchy boundary such that 0 < codim M ( C X) < 2. Case 2 non-polar Cauchy boundary: Letw(x,x + 1) = 2 x for all x N 0. It is easy to see that d is intrinsic for all α 1 2. Furthermore, since 1 x=0 w(x,x+1) < one can show that Cap( C X) > 0asinExample 5.3. Consequently, there exists a family of graphs with non-polar Cauchy boundary such that 0 < codim M ( C X) < 2. Acknowledgments M.K. enjoyed various inspiring discussion with Daniel Lenz and gratefully acknowledges the financial support from the German Research Foundation (DFG). R.K.W. thanks Józef Dodziuk for numerous insights and acknowledges the financial support of the FCT under project PTDC/MAT/101007/2008 and of the PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York. The authors are grateful to Ognjen Milatovic for a careful reading of the manuscript and suggestions. Appendix A. A Hopf Rinow type theorem Let (X,w,μ)be a weighted graph. A metric space (X, d) is said to be metrically complete if every Cauchy sequence converges to an element in X. A path (x n ) (finite or infinite) is called a geodesic with respect to a path metric d = d σ if d(x 0,x n ) = l σ ((x 0,...,x n )) for all n>0. A weighted graph (X,w,μ) with a path metric d = d σ is said to be geodesically complete if all infinite geodesics have infinite lengths, i.e., l σ ((x k )) = lim n l σ ((x 0,...,x n )) = for all infinite geodesics (x k ).

236 1574 X. Huang et al. / Journal of Functional Analysis 265 (2013) We prove the following Hopf Rinow type theorem. Theorem A.1. Let (X,w,μ) be a locally finite weighted graph and d be a path pseudo metric. Then, (X, d) is a discrete metric space. Moreover, the following are equivalent: (i) (X, d) is metrically complete. (ii) (X, d) is geodesically complete. (iii) Every distance ball is finite. (iv) Every bounded and closed set is compact. In particular, if (X, d) is complete, then for all x,y X there is a path (x 0,...,x n ) connecting x and y such that d σ (x, y) = l σ ((x 0,...,x n )). Remark A.2. (a) Anytime a path pseudo metric d induces the discrete topology on X the following implications hold: (iii) (iv) (i) (ii). This is the case if and only if inf y x σ(x,y) > 0 for all x X. In fact, (iv) (i) holds for general metric spaces. The stronger assumption of local finiteness is needed for the implications (ii) (i), (i) (iii) (or (iv)) and (ii) (iii) (or (iv)). See Example A.5 below. (b) A similar statement as (i) (iii) can also be found in [41]. We prove the theorem in several steps through the following lemmas. Lemma A.3. Let (X,w,μ) be a locally finite weighted graph and d be a path pseudo metric. Then, the following hold: (a) (X, d) is a discrete metric space. In particular, (X, d) is locally compact. (b) A set is compact in (X, d) if and only if it is finite. Proof. Local finiteness and the assumption σ(x,y) > 0forx y, imply that for all x X there is an r>0such that d(x,y) > r for all y X with y x. First, by the definition of d, wehave that for all x,z X with x y there is y x with d(x,y) d(x,z). Thus, d(x,y) = 0 implies x = y, therefore, d is a metric. Second, it yields that B r (x) ={x} and {x} is an open set which shows (a). From this we conclude that for any infinite set U the cover {{x} x U} has no finite subcover. The other direction of (b) is clear. The authors are grateful to Florentin Münch for a crucial idea in the proof of the following lemma. Lemma A.4. Let (X,w,μ) be a locally finite weighted graph and d be a path metric. Assume that B r is infinite for some r 0. Then, there exists an infinite geodesic of bounded length. Proof. Let o X be the center of the infinite ball B r of radius r and let d N be the natural graph distance. Let P n, n>0, be the set of finite paths (x 0,...,x k ) such that x 0 = o, x i x j for i j, d N (x k,o)= n and d N (x j,o) n for j = 0,...,k. Claim. Γ n ={γ P n γ geodesic with respect to d,l(γ) r} for all n>0.

237 X. Huang et al. / Journal of Functional Analysis 265 (2013) Proof. The set P n is finite by local finiteness of the graph and, thus, contains a minimal element γ = (x 0,...,x K ) with respect to the length l, i.e., for all γ P n we have l(γ ) l(γ). Then, γ is a geodesic: for every path (x 0,...,x M ) with x 0 = o and x M = x K,weletm {n,...,m} be such that (x 0,...,x m ) P n. By the minimality of γ we infer l (( x 0,...,x M)) l (( x 0,...,x m)) l(γ). It follows that γ is a geodesic. Clearly, l(γ) r, as otherwise B r {y X d N (y, o) n 1} which would imply the finiteness of B r by the local finiteness of the graph. Thus, γ Γ n which proves the claim. We inductively construct an infinite geodesic (x k ) with bounded length: We set x 0 = o. Since Γ n, there is a geodesic in Γ n for every n>0which contains x 0. Suppose we have constructed a geodesic (x 0,...,x m ) such that for all n m there is a geodesic in Γ n that has (x 0,...,x m ) as a subgeodesic. By local finiteness x m has finitely many neighbors. Thus, there must be a neighbor x m+1 of x m such that for infinitely many n the path (x 0,...,x m,x m+1 ) is a subpath of a geodesic in Γ n. However, a subpath of a geodesic is a geodesic. Thus, there is an infinite geodesic γ = (x k ) k 0 with l(γ) = lim n l((x 0,...,x n )) r as (x 0,...,x n ) Γ n for all n 0. Proof of Theorem A.1. The fact that (X, d) is a discrete metric space follows from Lemma A.3. We now turn to the proof of the equivalences. We start with (i) (ii). If there is a bounded infinite geodesic, then it is a Cauchy sequence. Since a geodesic is a path it is not eventually constant, thus it does not converge by discreteness. Hence, (X, d) is not metrically complete. To prove (ii) (iii) suppose that there is a distance ball that is infinite. By Lemma A.4 there is a bounded infinite geodesic and (X, d) is not geodesically complete. From Lemma A.3(b) we deduce (iii) (iv). Finally, we consider the direction (iv) (i). If every bounded and closed set is compact, then every closed distance ball is compact. Then, by Lemma A.3(b) every distance ball is finite and it follows that (X, d) is metrically complete. We finish this appendix by giving several (counter-)examples to show that some of the statements above fail to be true in the case of non-locally finite graphs. We present the examples with respect to the path metric with σ = σ 0 (see Example 2.1). Another example of this type can be found in [11, Example 14.1]. Example A.5. Let μ 1, σ = σ 0, and d = d σ. (1) A metrically and geodesically complete graph with non-compact distance balls. This example can be thought of as a star graph, where the rays are two subsequent edges. Let X = N 0 and let w be symmetric with w(0, 2n) = 1/2 n and w(2n 1, 2n) = 1 1/2 n for n N and w 0 otherwise. We have d(0, 2n) = 1forn N. Then, (X, d) is metrically (and geodesically) complete but B 1 (0) is not compact. (2) A non-locally compact graph. This example can be thought of as a star graph where the rays are copies of N whose lengths become shorter. Let X = N 2 0 and let w be symmetric with w((0, 0), (m, 0)) = 1/2m for m N and w((m, n 1), (m, n)) = 2 2(m+n) /5form, n N and w 0 otherwise. Then, Deg((0, 0)) = 1, Deg((m, 0)) = 1/2 m + 2 2(m+1) /5 and Deg((m, n)) = 2 2(m+n) for m, n N. Hence, we have d((m, n 1), (m, n)) = 2 (m+n) for m, n N and 1/2 m+1 d((0, 0), (m, n))

238 1576 X. Huang et al. / Journal of Functional Analysis 265 (2013) /2 m+1. For a ball B r ((m, n)), m, n 0, r>0, denote by U r ((m, n)) its interior. Now for ε>0, {U ε/2 ((0, 0))} {U 1/2 (m+n+1)((m, n)) m 1,n 0} is an open cover of B ε ((0, 0)) with no finite subcover. (3) A non-hausdorff space. This example can be thought as two vertices which are connected by infinitely many paths that become shorter. Let X = N 0 { }and let w be symmetric with w(0, 2n) = w(, 2n) = 1/2 n and w(2n 1, 2n) = 2 2n and w(n,m) = 0 all other m, n N 0. Then, σ(0, 2n) = σ(, 2n) 1/2 n. Hence, d(0, ) = 0. (4) An infinite ball and non-discreteness. This example is a modification of (1). Let X = N 0 and let w be symmetric with w(0, 2n) = 1/2 n and w(2n 1, 2n) = 2 n and w(n,m) = 0 all other m, n N 0. Then, every d-ball about 0 is compact but it contains infinitely many vertices. Moreover, the vertices x n = 2n converge to x = 0 with respect to d. (This is, in particular, a counterexample to Lemma A.3 for non-locally finite graphs.) (5) A geodesically complete graph which is not metrically complete. This example is an extension of (3) and can be thought as a line graph where between each two points on the line there are infinitely many line segments that become shorter. Let X = N 2 0 and let w be symmetric with w((m, 0), (m, 2n)) = 1/2n = w((m + 1, 0), (m, 2n)) and w((m, 2n), (m, 2n 1)) = 2 2(m+1) 3/2 n for m N 0,n N and w 0 otherwise. It follows that Deg((m, 2n)) = 2 2(m+1) 1/2 n implying that d((m, 0), (m + 1, 0)) = 1/2 m. Thus (x m ) = ((m, 0)) is a Cauchy sequence which does not converge. On the other hand, the space is geodesically complete as there are no infinite geodesics. References [1] J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds, in: Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, in: Proc. Sympos. Pure Math., vol. XXXVI, American Mathematical Society, Providence, RI, 1980, pp [2] P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973) [3] Y. Colin de Verdière, Pseudo-laplaciens. I, Ann. Inst. Fourier 32 (3) (1982) [4] Y. Colin de Verdière, N. Torki-Hamza, F. Truc, Essential self-adjointness for combinatorial Schrödinger operators II Metrically non complete graphs, Math. Phys. Anal. Geom. 14 (1) (2011) [5] Y. Colin de Verdière, N. Torki-Hamza, F. Truc, Essential self-adjointness for combinatorial Schrödinger operators III Magnetic fields, Ann. Fac. Sci. Toulouse Math. (6) 20 (3) (2011) [6] J. Dodziuk, Elliptic Operators on Infinite Graphs, Analysis, Geometry and Topology of Elliptic Operators, World Sci. Publ., Hackensack, NJ, 2006, pp [7] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 2nd ed., John Wiley & Sons Inc., Hoboken, NJ, [8] M. Folz, Gaussian upper bounds for heat kernels of continuous time simple random walks, Elec. J. Prob. 16 (2011) [9] M. Folz, Volume growth and stochastic completeness of graphs, Trans. Amer. Math. Soc. (2013), in press. [10] M. Folz, Volume growth and spectrum for general graph Laplacians, Math. Z. (2013), in press. [11] R.L. Frank, D. Lenz, D. Wingert, Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory, J. Funct. Anal. (2013), in press. [12] K. Friedrichs, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109 (1) (1934) [13] M. Fukushima, Regular representations of Dirichlet spaces, Trans. Amer. Math. Soc. 155 (1971) [14] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, [15] M.P. Gaffney, The harmonic operator for exterior differential forms, Proc. Nat. Acad. Sci. USA 37 (1951)

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241 CHAPTER 10 F. Bauer, M. Keller, R. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, to appear in Journal of the European Mathematical Society. 239

242 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS FRANK BAUER, MATTHIAS KELLER, AND RADOS LAW K. WOJCIECHOWSKI Abstract. We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded. 1. Introduction In 1984 Dodziuk [9] proved a lower bound on the spectrum of the Laplacian on infinite graphs in terms of an isoperimetric constant. Dodziuk s bound is an analogue of Cheeger s inequality for manifolds [6] except for the fact that Dodziuk s estimate also contains an upper bound for the vertex degrees in the denominator. In a later paper [12] Dodziuk and Kendall expressed that it would be desirable to have an estimate without the rather unnatural vertex degree bound. They overcame this problem in [12] by considering the normalized Laplace operator, which is always a bounded operator, instead. However, the original problem of finding a lower bound on the spectrum of unbounded graph Laplace operators that only depends on an isoperimetric constant remained open until today. In this paper, we solve this problem by using the concept of intrinsic metrics. More precisely, for a given weighted Laplacian, we use an intrinsic metric to redefine the boundary measure of a set. This leads to a modified definition of the isoperimetric constant for which we obtain a lower bound on the spectrum that depends solely on the constant. These estimates hold for all weighted Laplacians (including bounded and unbounded Laplace operators). The strategy of proof is not surprising, as it does not differ much from the one of [9, 12]. However, the main contribution of this note is to provide the right definition of an isoperimetric constant to solve the open problem mentioned above. Date: March 17,

243 2 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI To this day, there is a vital interest in estimates of isoperimetric constants and in Cheeger-type inequalities. For example, rather classical estimates for isoperimetric constants in terms of the vertex degree can be found in [2, 11, 43, 44]; for relations to random walks, see [21, 48]. While, for regular planar tessellations, isoperimetric constants can be computed explicitly [26, 28]; for arbitrary planar tessellations there are estimates in terms of curvature [27, 35, 40, 47]. For Cheeger inequalities on simplicial complexes, there is recent work found in [45]; for general weighted graphs, see [10, 37]. Moreover, Cheeger estimates for the bottom of the essential spectrum and criteria for discreteness of spectrum are given in [19, 34, 49, 50]. Upper bounds for the top of the (essential) spectrum and another criterium for the concentration of the essential spectrum in terms of the dual Cheeger constant are given in [3]. Finally, let us mention works connecting discrete and continuous Cheeger estimates [1, 8, 33, 41]. The paper is structured as follows. The set up is introduced in the next section. The Cheeger inequalities are presented and proven in Section 3. Moreover, upper bounds are discussed. A technique to incorporate non-negative potentials into the estimate is discussed in Section 4. Section 5 is dedicated to relating the exponential volume growth of a graph to the isoperimetric constant via upper bounds while lower bounds on the isoperimetric constant in the flavor of curvature are presented in Section 6. These lower bounds allow us to give examples where our estimate yields better results than all estimates known before. 2. The set up 2.1. Graphs. Let X be a countably infinite set equipped with the discrete topology. A function m : X (0, ) gives a Radon measure on X of full support via m(a) = x A m(x) for A X, so that (X, m) becomes a discrete measure space. A graph over (X, m) is a symmetric function b : X X [0, ) with zero diagonal that satisfies b(x, y) < for x X. y X We can think of x and y as neighbors, i.e, being connected by an edge, if b(x, y) > 0 and we write x y. In this case, b(x, y) is the strength of the bond interaction between x and y. For convenience we assume that there are no isolated vertices, i.e., every vertex has a neighbor. We call b locally finite if each vertex has only finitely many neighbors.

244 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS 3 The measure n : X (0, ) given by n(x) = y X b(x, y) for x X. plays a distinguished role in the proof of classical Cheeger inequalities. In the case when b : X X {0, 1}, n(x) gives the number of neighbors of a vertex x Intrinsic metrics. We call a pseudo metric d for a graph b on (X, m) an intrinsic metric if b(x, y)d(x, y) 2 m(x) for all x X. y X The concept of intrinsic metrics was first studied systematically by Sturm [46] for strongly local regular Dirichlet forms and it was generalized to all regular Dirichlet forms by Frank/Lenz/Wingert in [17]. By [17, Lemma 4.7, Theorem 7.3] it can be seen that our definition coincides with the one of [17]. A possible choice for d is the path pseudo metric induced by the edge weights w(x, y) = ((m/n)(x) (m/n)(y)) 1 2, for x y, see e.g. [30]. Moreover, the natural graph metric (i.e., the path metric with weights w(x, y) = 1 for x y) is intrinsic if m n. Intrinsic metrics for graphs were recently discovered independently in various contexts, see e.g. [14, 15, 16, 23, 25, 30, 31, 32, 42], where certain variations of the concept also go under the name adapted metrics Isoperimetric constant. In this section we use the concept of intrinsic metrics to give a refined definition of the isoperimetric constant. As it turns out, this novel isoperimetric constant is more suitable than the classical one if n m. Let W X. We define the boundary W of W by W = {(x, y) W X \ W b(x, y) > 0}. For a given intrinsic metric d we set the measure of the boundary as W = b(x, y)d(x, y). (x,y) W Note that W < for finite W X by the Cauchy-Schwarz inequality and the assumption that y b(x, y) <. We define the isoperimetric constant or Cheeger constant α(u) = α d,m (U) for U X as α(u) = inf W Ufinite W m(w ).

245 4 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI If U = X, we write α = α(x). For b : X X {0, 1} and d the natural graph metric the measure of the boundary W is number of edges leaving W. If additionally m = n, then our definition of α coincides with the classical one from [12] Graph Laplacians. Denote by C c (X) the space of real valued functions on X with compact support. Let l 2 (X, m) be the space of square summable real valued functions on X with respect to the measure m which comes equipped with the scalar product u, v = x X u(x)v(x)m(x) and the norm u = u m = u, u 1 2. Let the form Q = Q b with domain D be given by Q(u) = 1 2 x,y X b(x, y)(u(x) u(y)) 2, D = C c (X) Q, where Q = (Q( ) + 2 ) 1 2. The form Q defines a regular Dirichlet form on l 2 (X, m), see [20, 36]. The corresponding positive selfadjoint operator L can be seen to act as Lf(x) = 1 m(x) b(x, y)(f(x) f(y)), y X (cf. [36, Theorem 9]). Let L be the extension of L to F = {f : X R y X b(x, y) f(y) < for all x X}. We have C c(x) D(L) if (and only if) LC c (X) l 2 (X, m), see [36, Theorem 6]. In particular, this can easily seen to be the case if the graph is locally finite or if inf x X m(x) > 0. Note that L becomes a bounded operator if and only if Cm n for some C > 0 (cf. [24, Theorem 9.3]). In particular, if m = n, then L is referred to as the normalized Laplacian. We denote the bottom of the spectrum σ(l) and the essential spectrum σ ess (L) of L by λ 0 (L) = inf σ(l) and λ ess 0 (L) = inf σ ess (L). 3. Cheeger inequalities Let b be a graph over (X, m) and d be an intrinsic metric. In this section we prove the main results of the paper.

246 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS Main results. Theorem 3.1. λ 0 (L) α2 2. Remark. (a) A similar statement can be proven for weighted Laplacians on finite graphs for the first non-zero eigenvalue. In this case, the infimum in the definition of the isoperimetric constant has to be taken over all sets that have at most half of the measure of the whole graph. The main part of the proof works similarly, for details of the adaption to the finite graph case, see [7, proof of Theorem 2.2]. (b) We also obtain a similar statement for the self adjoint operator that is related to the maximal form (cf. Section 5) which is discussed under the name Neumann Laplacian in [24]. Here one has to take the infimum in the definition of the isoperimetric constant over all sets of finite measure. With this choice, all of our proofs work analogously. Under the additional assumption that L is bounded with operator norm 1, we recover the classical Cheeger inequality from [19, 43] which can be seen to be stronger than the one of [12] by the Taylor expansion. We say that d 1 (respectively d 1) for neighbors if d(x, y) 1 (respectively d(x, y) 1) for all x y. Theorem 3.2. If m n and d 1 or d 1 for neighbors, then λ 0 (L) 1 1 α 2. In order to estimate the essential spectrum let the isoperimetric constant at infinity be given by α = sup α(x \ K), K Xfinite which coincides with the one of [37] in the case of the natural graph metric and with the one of [19, 34] if additionally b : X X {0, 1} and m = n. Note that the assumptions of the following theorem are in particular fulfilled if the graph is locally finite or if inf x X m(x) > 0. Theorem 3.3. Assume C c (X) D(L). Then, λ ess 0 (L) α Co-area formulae. Among the key ingredients for the proof are the following well-known area and co-area formulae. For example, they are already found in [37], see also [22]. We include a short proof for the sake of convenience. Let l 1 (X, m) = {f : X R x X f(x) m(x) < }.

247 6 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI Lemma 3.4. Let f l 1 (X, m), f 0 and Ω t := {x X f(x) > t}. Then, 1 2 x,y X b(x, y)d(x, y) f(x) f(y) = 0 Ω t dt, where the value on both sides of the equation is allowed, and f(x)m(x) = x X 0 m(ω t )dt. Proof. For x, y X, x y with f(x) f(y), let the interval I x,y be given by I x,y := [f(x) f(y), f(x) f(y)) and let I x,y = f(x) f(y) be the length of I x,y. Then, (x, y) Ω t if and only if t I x,y. Hence, by Fubini s theorem, 0 Ω t dt = 1 2 = 1 2 = x,y X x,y X x,y X b(x, y)d(x, y)1 Ix,y (t)dt b(x, y)d(x, y) 0 1 Ix,y (t)dt b(x, y)d(x, y) f(x) f(y). Note that x Ω t if and only if 1 (t, ) (f(x)) = 1. Again, by Fubini s theorem, 0 m(ω t )dt = 3.3. Form estimates. 0 m(x)1 (t, ) (f(x))dt x X = x X m(x) 0 1 (t, ) (f(x))dt = x X m(x)f(x). Lemma 3.5. For U X and u D with support in U and u m = 1 Q(u) α(u)2. 2 Moreover, if m n and d 1 or d 1 for neighbors, then Q(u) 2 2Q(u) + α(u) 2 0.

248 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS 7 Proof. Let u C c (X). We calculate, using the co-area formulae above with f = u 2, α u 2 m = α = x,y X Q(u) 1 2 Q(u) 1 2 m(ω t )dt 0 Ω t dt b(x, y)d(x, y) u 2 (x) u 2 (y) ( 1 2 x,y X b(x, y)d(x, y) 2 (u(x) + u(y)) 2 ) 1 2 ( 2 u(x) ) 1 2 b(x, y)d(x, y) 2 2 x X y X Q(u) 1 2 u m, where the final estimate follows from the intrinsic metric property. The second statement follows if we use in the above estimates 1 b(x, y)d(x, y) 2 (u(x) + u(y)) 2 2 x,y X = 2 x,y X 2 u 2 m Q(u), b(x, y)d(x, y) 2 u(x) x,y X b(x, y)d(x, y) 2 (u(x) u(y)) 2 where we distinguish the cases d 1 and d 1: For the first case we use that d is intrinsic and that d(x, y) 2 1. For the second case, we estimate d(x, y) 1 in the first line and then use n m. The statement follows by the density of C c (X) in D Proof of the theorems. Proof of Theorem 3.1 and Theorem 3.2. By virtue of Lemma 3.5, the statements follow by the variational characterization of λ 0 via the Rayleigh Ritz quotient: λ 0 = inf u D, u =1 Q(u). Proof of Theorem 3.3. Let Q U, U X, be the restriction of Q to C c (U) Q and L U be the corresponding operator. Note that Q U = Q on C c (U). The assumption C c (X) D(L) clearly implies LC c (X) l 2 (X, m) which is equivalent to the fact that functions y b(x, y)/m(y) are in l 2 (X, m) for all x X, see [36, Proposition 3.3]. This implies that for any finite set K X the operator L X\K is a compact perturbation of L. Thus, from Lemma 3.5 we conclude λ ess 0 (L) = λ ess α(x \ K)2 0 (L X\K ) λ 0 (L X\K ) = inf Q(u). u C c(x\k), u =1 2

249 8 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI This implies the statement of Theorem Upper bounds for the bottom of spectrum. In this section we show an upper bound for λ 0 (L) by α as in [9, 11, 12] for uniformly discrete metric spaces. Theorem 3.6. Let d be an intrinsic metric such that (X, d) is uniformly discrete with lower bound δ > 0. Then, λ 0 (L) α/δ. Proof. By assumption we have d δ > 0 away from the diagonal. It follows that W δ (x,y) W b(x, y) = δq(1 W ) for all W X finite. By the inequality δλ 0 (L) δq(1 W )/ 1 W 2 W /m(w ), we conclude the statement. The example below shows that, in general, there is no upper bound by α only. Example 3.7. Let b 0 : X X {0, 1} be a k-regular rooted tree with root x 0 X (that is, each vertex has k forward neighbors). Furthermore, let b 1 : X X {0, 1} be such that b 1 (x, y) = 1 if and only if x and y have the same distance to x 0 with respect to the natural graph distance in b 0, and b 1 (x, y) = 0 otherwise. Now, let b = b 0 + b 1, m 1 and let d be given by the path metric with weights w(x, y) = (n(x) n(y)) 1 2 for x y. Then, α = α d,m = 0 which can be seen by B r /m(b r ) k (r 1)/2 0 as r, where B r is the set of vertices that have distance less or equal r to with respect to the natural graph metric. On the other hand, by [39, Theorem 2] the heat kernel p t (x 0, ) of the graph b equals the corresponding heat kernel on the k-regular tree b 0. Hence, by a Li type theorem, see [24, Theorem 8.1] or [38, Corollary 5.6], we get 1 log p t t(x 0, y) λ 0 (L) = (k k) for any y X as t (see also [39, Corollary 6.7]). As α = 0, this shows that α can yield no upper bound without further assumptions. Nevertheless, the example does not exclude the possibility that there is a different intrinsic metric which might yield a reasonable upper bound. Hence, one might ask whether there exist examples for which every intrinsic metric fails to give an upper bound or, otherwise, if one can always find an intrinsic metric that yields an upper bound. 4. Potentials In this section we briefly discuss how the strategy proposed in [37] to incorporate potentials into the inequalities can be applied to the new

250 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS 9 definition of the Cheeger constant. This yields a Cheeger estimate for all regular Dirichlet forms on discrete sets (cf. [36, Theorem 7]). Let b be a graph over a discrete measure space (X, m). Furthermore, let c : X [0, ) be a potential and define Q b,c (u) = 1 b(x, y)(u(x) u(y)) 2 + c(x)u(x) 2 2 x,y X x X on D(Q b,c ) = C c (X) Q b,c and let L b,c be the corresponding operator. Let (X, b, m ) be a copy of (X, b, m). Let Ẋ = X X, ṁ : Ẋ (0, ) such that ṁ X = m, ṁ X = m and let ḃ : Ẋ Ẋ [0, ) be given by ḃ X X = b, ḃ X X = b, ḃ(x, x ) = c(x) = c (x ) for corresponding vertices x X and x X and ḃ 0 otherwise. Then, the restriction Qḃ,X of the form Qḃ on l 2 (Ẋ, ṁ) to D(Q ḃ,x ) = C c(x) Qḃ satisfies D(Q b,c ) = D(Qḃ,X ) and Q b,c = Qḃ,X. Let d : X X [0, ) be an intrinsic metric for b over (X, m) and assume there is a function δ : X [0, ) such that b(x, y)d(x, y) 2 + c(x)δ(x) 2 m(x) for all x X. y X Example 4.1. (1) For a given intrinsic metric d a possible choice for the function δ is δ(x) = ((m(x) y X b(x, y)d(x, y)2 )/c(x)) 1 2 if c(x) > 0 and 0 otherwise. (2) Choose d as the path metric induced by the edge weights w(x, y) = (( m )(x) ( m )(y)) 1 2 for x y and δ as in (1). If c > 0, then δ > 0. n+c n+c We next define d. Since we are only interested in the subgraph X of Ẋ, we do not need to extend d to all of Ẋ but only set d X X = d and d(x, x ) = δ(x) for the corresponding vertex x X of x. Defining α(x) = α d,m (X) by α(x) = inf W Xfinite W d m(w ) with W d = (x,y) W b(x, y)d(x, y) + x W c(x)δ(x) implies that α(x) = αd,ṁ (X), where the right hand side is the Cheeger constant of the subgraph X Ẋ as in Section 2.3. Hence, we get λ 0 (L b,c ) α(x)2 2 by Lemma 3.5 and the arguments from the proof of Theorem 3.1.

251 10 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI 5. Upper bounds by volume growth In this section we relate the isoperimetric constant to the exponential volume growth of the graph. Let b be a graph over a discrete measure space (X, m) and let d be an intrinsic metric. We let B r (x) = {y X d(x, y) r} and define the exponential volume growth µ = µ d,m by µ = lim inf r inf 1 x X r log m(b r(x)) m(b 1 (x)). Other than for the classical notions of isoperimetric constants and exponential volume growth on graphs (see [5, 11, 18, 43]), it is, geometrically, not obvious that α = α d,m and µ = µ d,m can be related. However, given a Brooks-type theorem, the proof is rather immediate. Therefore, let the maximal form domain be given by D max := {u l 2 (X, m) Q max (u) = 1 b(x, y)(u(x) u(y)) 2 < }. 2 x,y X Theorem 5.1. If D = D max, then 2α µ. In particular, this holds if one of the following assumptions is satisfied: (a) The graph b is locally finite and d is an intrinsic path metric such that (X, d) is metrically complete. (b) Every infinite path of vertices has infinite measure. Proof. Under the assumption D = D max we have λ 0 (L) µ 2 /8 by [25, Theorem 4.1]. (Note that the 8 in the denominator as opposed to the 4 found in [25] is explained in [25, Remark 3].) Thus, the statement follows by Theorem 3.1. Note that, by [32, Theorem 2] and [24, Corollary 6.3], (a) implies D = D max and, by [36, Theorem 6], (b) implies D = D max. 6. Lower bounds by curvature In this section we give a lower bound on the isoperimetric constant by a quantity that is sometimes interpreted as curvature [11, 29, 39]. Let b be a graph over (X, m) and let d be an intrinsic metric The lower bound. We fix an orientation on a subset of the edges, that is, we choose E +, E X X with E + E = such that (x, y) E + if and only if (y, x) E. We define the curvature with respect to this orientation by K : X R K(x) = 1 ( ) b(x, y)d(x, y) b(x, y)d(x, y) m(x) (x,y) E (x,y) E + Let us give an example for a choice of E ±.

252 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS 11 Example 6.1. Let b take values in {0, 1}, m be the vertex degree function n, and d be the natural graph metric. For some fixed vertex x 0 X, let S r be the spheres with respect to d around x 0 and x = r for x S r. We choose E ± such that outward (inward) oriented edges are in E + (E ), i.e., (x, y) E +, (y, x) E if x S r 1, y S r for some r and x y. Then K(x) = (n (x) n + (x))/n(x), where n ± (x) = #{y S x ±1 y x} and #A denotes the cardinality of A. The following theorem is an analogue to [11, Lemma 1.15] and [13, Proposition 3.3] which was also used in [49, 50] to estimate the bottom of the essential spectrum. Theorem 6.2. If K k 0, then α k. Proof. Let W be a finite set and denote by 1 W the corresponding characteristic function. Furthermore, let σ(x, y) = ±d(x, y) for (x, y) E ± and zero otherwise. We calculate directly km(w ) K(x)m(x) = 1 W (x) b(x, y)σ(x, y) x W x X y X = 1 ( 1 W (x) b(x, y)σ(x, y) 1 W (y) ) b(x, y)σ(x, y) 2 x X y X y X x X 1 b(x, y)d(x, y) 1 W (x) 1 W (y) = W, 2 x,y X where we used y b(x, y)d(x, y) < and the antisymmetry of σ in the second step. This finishes the proof Example of antitrees. In the final subsection we give an example of an antitree for which Theorem 6.2 together with Theorem 3.1 yields a better estimate than the estimates known before. Recently, antitrees received some attention as they provide examples of graphs of polynomial volume growth (with respect to the natural graph metric) that are stochastically incomplete and have a spectral gap, see [4, 23, 25, 31, 39, 51]. For a given graph b : X X {0, 1} with root x 0 X and measure m 1, let S r be the vertices that have natural graph distance r to x 0 as above. We call a graph an antitree if every vertex in S r is connected to all vertices in S r+1 S r 1 and to none in S r. In [25], it is shown that λ 0 (L) = 0 whenever lim r log #S r / log r < 2. It remains open by this result what happens in the case of an antitree with #S r 1 = r 2. The classical Cheeger constant α classical = α 1,n for the normalized Laplacian with the natural graph metric which is given

253 12 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI as the infimum over # W/n(W ) (with W X finite) is zero. This can be easily checked by choosing distance balls B r = r j=0 S j as test sets W. Hence, the estimate λ 0 (L) (1 1 α 2 ) inf x X n(x) with α = α 1,n found in [34] is trivial. Likewise, the estimates presented in [39] and [50], which uses an unweighted curvature, also give zero as a lower bound for the bottom of the spectrum in this case. By Theorem 6.2 we obtain a positive estimate for the Cheeger constant α = α d,1 with the path metric induced by the edge weights w(x, y) = (n(x) n(y)) 1 2, x y for the antitree with #S r 1 = r 2. We pick E ± as in Example 6.1 above and obtain a positive lower bound for K. In particular, Theorem 6.2 shows that α > 0 and, thus, λ 0 (L) > 0 by Theorem 3.1 for the antitree satisfying #S r 1 = r 2. Acknowledgements. F.B. thanks Jürgen Jost for introducing him to the topic and for many stimulating discussions during the last years. The research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme (FP7/ ) / ERC grant agreement n and the Alexander von Humboldt foundation. M.K. thanks Daniel Lenz for sharing generously his knowledge about intrinsic metrics and he enjoyed vivid discussions with Markus Seidel, Fabian Schwarzenberger and Martin Tautenhahn. M.K. also gratefully acknowledges the financial support from the German Research Foundation (DFG). R.K. thanks Józef Dodziuk for introducing him to Cheeger constants and for numerous inspiring discussions. Support for this project was provided by PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York. References [1] E. Arias-Castro, B. Pelletier, P. Pudlo, The normalized graph cut and Cheeger constant: from discrete to continuous, Adv. in Appl. Probab. 44 (2012), no. 4, [2] N. Alon, V. D. Milman, λ 1, isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), no. 1, [3] F. Bauer, B. Hua, J. Jost, The dual Cheeger constant and spectra of infinite graphs, preprint 2012, arxiv: [4] J. Breuer, M. Keller, Spectral analysis of certain spherically homogeneous graphs, to appear in: Oper. Matrices. [5] N. L. Biggs, B. Mohar, J. Shawe-Taylor, The spectral radius of infinite graphs, Bull. London Math. Soc. 20 (1988), no. 2, [6] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, N. J., 1970, [7] F. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.

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255 14 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI [29] X. Huang, Stochastic incompleteness for graphs and weak Omori-Yau maximum principle, J. Math. Anal. Appl. 379 (2011), no. 2, [30] X. Huang, On stochastic completeness of weighted graphs, Ph.D. thesis, [31] X. Huang, On uniqueness class for a heat equation on graphs, J. Math. Anal. Appl. 393 (2012), no. 2, [32] X. Huang, M. Keller, J. Masamune, R. K. Wojciechowski, A note on self-adjoint extensions of the Laplacian on weighted graphs, preprint 2012, arxiv: [33] M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, [34] M. Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346 (2010), no. 1, [35] M. Keller, Curvature, geometry and spectral properties of planar graphs, Discrete Comput. Geom. 46 (2011), no. 3, [36] M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math. 666 (2012), [37] M. Keller, D. Lenz, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5 (2010), no. 4, [38] M. Keller, D. Lenz, H. Vogt, R. Wojciechowski, Note on basic features of large time behaviour of heat kernels, preprint 2011, arxiv: [39] M. Keller, D. Lenz, R. K. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, to appear in: Math. Z. [40] M. Keller, N. Peyerimhoff, Cheeger constants, growth and spectrum of locally tessellating planar graphs, Math. Z. 268 (2011), no. 3-4, [41] S. Markvorsen, S. McGuninness, C. Thomassen, Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces, Proc. London Math. Soc. (3) 64 (1992), no. 1, [42] J. Masamune and T. Uemura, Conservation property of symmetric jump processes, Ann. Inst. Henri. Poincaré Probab. Stat. 47 (2011), no. 3, [43] B. Mohar, Isoperimetics inequalities, growth and the spectrum of graphs, Linear Algebra Appl. 103 (1988), [44] B. Mohar, Some relations between analytic and geometric properties of infinite graphs, Discrete Math. 95 (1991), no. 1-3, [45] O. Parzanchevski, R. Rosenthal and R.J. Tessler, Isoperimetric inequalities in simplicial complexes, preprint 2012, arxiv: [46] K.-T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p -Liouville properties, J. Reine Angew. Math. 456 (1994), [47] W. Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, [48] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, [49] R. K. Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.) City University of New York. [50] R. K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J. 58 (2009), no. 3,

256 CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS 15 [51] R. K. Wojciechowski, Stochastically incomplete manifolds and graphs, Random walks, boundaries and spectra, Progr. Prob., vol. 64, Birkhäuser Verlag, Basel, 2011, Frank Bauer, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA and Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, Leipzig, Germany address: Matthias Keller, Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel and Mathematisches Institut Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, Jena, Germany address: Rados law K. Wojciechowski, York College of the City University of New York, Jamaica, NY 11451, USA address:

257 CHAPTER 11 S. Haeseler, M. Keller, R. Wojciechowski, Volume growth and bounds for the essential spectrum for Dirichlet forms, Journal of the London Mathematical Society 88 (2013),

258 J. London Math. Soc. (2) 88 (2013) C 2013 London Mathematical Society doi: /jlms/jdt029 Volume growth and bounds for the essential spectrum for Dirichlet forms Sebastian Haeseler, Matthias Keller and Rados law K. Wojciechowski Abstract We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases, we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric), we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth. 1. Introduction and main results In 1981, Brooks proved that the bottom of the essential spectrum of the Laplace Beltrami operator on a complete non-compact Riemannian manifold with infinite measure can be bounded by the exponential volume growth rate of the manifold [2]. Following this, similar results were proved in various contexts; see [3, 6, 12, 13, 22 25, 28]. Very recently, it was shown in [19] that such a result fails to be true in the case of the (non-normalized) graph Laplacian when the volume is measured with respect to the natural graph distance. Indeed, there are graphs of cubic polynomial volume growth that have positive bottom of the spectrum and slightly more than cubic growth already allows for a purely discrete spectrum. This suggests that one should look at other candidates for a metric on a graph. In this paper, we work in the context of regular Dirichlet forms (without killing term) and use the corresponding concept of an intrinsic metric (see [5, 28]) to prove a Brooks-type theorem. The purpose of this approach is threefold. First, we provide a setup which includes all known examples (and various others, for example, quantum graphs) and give a unified treatment. Additionally, our estimates are better than most of the previous results (by replacing a lim sup by a lim inf for the essential spectrum). Second, our method of proof seems to be much clearer and simpler than most of the previous works. Finally, graph Laplacians are now included and the disparity discussed above is resolved by considering suitable metrics. As an application, we can now prove that the examples found in [19] for Laplacians on graphs do indeed give the borderline for positive bottom of the spectrum. In particular, for the natural graph distance, the threshold for zero bottom of the essential spectrum and the discreteness of the spectrum lies at cubic growth. Let X be a locally compact separable metric space and m be a positive Radon measure of full support. Let E be a closed, symmetric, non-negative form on the Hilbert space L 2 (X, m) of real-valued square-integrable functions with domain D. We assume that E is a regular Dirichlet Received 3 July 2012; revised 9 January 2013; published online 26 September Mathematics Subject Classification 47D07 (primary), 58J50, 35P05, 39A12 (secondary). The research of Rados law K. Wojciechowski was partially sponsored by the Fundação para a Ciência e a Tecnologia through project PTDC/MAT/101007/2008 and by PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York. Matthias Keller appreciates the financial support of the German Science Foundation (DFG).

259 884 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI form without killing term (for background on Dirichlet forms see [7]; more details are given in Subsection 2.1). Let L be the positive self-adjoint operator arising from E. Define λ 0 (L) :=infσ(l) and λ ess 0 (L) :=infσ ess (L), where σ ess (L) denotes the essential spectrum of L. We let ρ be an intrinsic metric in the sense of [5]. (Note that an intrinsic metric is, in general, a pseudometric.) For x 0 X and r 0, we define the distance ball B r = B r (x 0 )={x X ρ(x, x 0 ) r}. Let the exponential volume growth be defined as μ = lim inf r 1 r log m(b r(x 0 )). Note that, in contrast to previous works on manifolds [2] and graphs [6], we consider a lim inf here, rather than a lim sup. If ρ takes values in [0, ), then X = r B r(x 0 ). In this case, μ does not depend on the particular choice of x 0. There is another constant first introduced in [28] which we call the minimal exponential volume growth and which is defined as μ = lim inf r In this paper, we prove the following theorem. 1 r inf x X log m(b r(x)) m(b 1 (x)). Theorem 1.1. Let L be the positive self-adjoint operator arising from a regular Dirichlet form E without killing term and let ρ be an intrinsic metric such that all distance balls are compact. Then, λ 0 (L) μ2 4. If, additionally, m( r B r(x 0 )) = for some x 0, then λ ess 0 (L) μ2 4. This has the following immediate corollary. The corollary has various consequences, for example, the exponential instability of the semigroup (e tl ) t 0 on L p (X, m), p [1, ]; see [28, Corollary 2]. Corollary 1.2. Suppose that (X, ρ) is of subexponential growth, that is, μ =0 (respectively, μ =0). Then, λ 0 (L) =0(respectively, λ ess 0 (L) =0). Remark 1.3. (a) Let us discuss Theorem 1.1 in the perspective of the present literature: for the Laplace Beltrami operator on a Riemannian manifold an estimate for λ ess 0 can be found in [2]; see also [12]. For strongly local Dirichlet forms, the statement for λ 0 is proved in [28] and for λ ess 0 in [24]; see also [22, 23] for more subtle estimates involving λ 0 in the case of manifolds. For non-local operators, such results were known only for normalized Laplacians on graphs; see [3, 6, 13, 25]. These operators are of a very special form, in particular, they are always bounded. For unbounded Laplacians on graphs, the conclusions of the theorem do not hold if one considers volume with respect to the natural graph metric; see [19]. However, by Frank, Lenz and Wingert [5] (see also [9]), there is now a suitable notion of an intrinsic metric for non-local forms. Let us stress that our result covers the results in [2, 3, 6, 25, 28]. Results of the

260 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 885 type found in [12, 13] could certainly also be obtained with slightly more technical effort which we avoid here for the sake of the clarity of presentation. (b) Despite the fact that our result is much more general, we have a unified method of proof for the bounds on the spectrum and the essential spectrum. For the essential spectrum, the proof is significantly simpler than the one of [2, 6] as we avoid a cutoff procedure by using test functions which converge weakly to zero. (c) Indeed, we prove a slightly more general result than above for non-local forms in Subsection 3.2. In particular, for some special cases we prove much better estimates and recover the results of [3, 6, 25] in Corollary 4.6 in Subsection 4.1. (d) If we assume that ρ takes values in [0, ), then we can clearly replace the assumption that m( r B r(x 0 )) = with m(x) =. The case when m(x) < is notably different; see [8] for more details. (e) If inf x X m(b 1 (x)) > 0, then one can also show λ ess 0 (L) μ 2 /4; see Remark 2.3 in Subsection 2.3. (f) Our result deals exclusively with Dirichlet forms with vanishing killing term. The major challenge in the case of a non-vanishing killing term is to give a proper definition of volume which incorporates the killing term. We shortly discuss a strategy for approaching this case: we need a positive generalized harmonic function u, thatis,e(u, ϕ) = 0 for all ϕ D, where u is assumed to be locally in the domain of E (this space is introduced in [5]asDloc ). Such a function exists in many settings (see, for example, [3, 11, 21]); the result guaranteeing the existence of such a function is often referred to as an Allegretto Piepenbrink-type theorem. Then, by a ground state representation (see [5, Theorem 10.1]), one obtains a form E u with vanishing killing term such that E = E u on the intersection of their domains. Now we can apply the methods above for E u to derive the result for E. However, as shown in [11], there are examples of non-locally finite weighted graphs that do not have such a generalized harmonic function. Therefore, it would be interesting to find sufficient conditions under which the approach above could be carried out. Let us highlight one of the applications of our results for graphs. Let Δ be the graph Laplacian on l 2 (X) acting as Δϕ(x) = y x(ϕ(x) ϕ(y)) (for more details, see Subsections 4.1 and 4.2). Moreover, let Br d,forr 0, be balls with respect to the natural graph distance d defined as the smallest number of edges in a path between two vertices. It has to be stressed that this metric is not an intrinsic metric for Δ. However, we will show in Theorem 4.7 that, if the growth of the balls Br d is smaller than r 3 ε for any ε>0, then λ 0 (Δ) = λ ess 0 (Δ) = 0 and, if it is less than r 3, then λ ess 0 (Δ) <. We demonstrate by examples that this result is sharp; see Subsection 4.2. The paper is structured as follows. In Section 2, we recall some basic facts about Dirichlet forms and intrinsic metrics. Moreover, we give a bound on the bottom of the essential spectrum via weak null sequences and introduce the test functions. In Section 3, we prove the crucial estimate for the strongly local and the non-local parts of the Dirichlet form and prove the main theorem. In Section 4, we discuss the result for weighted graphs and prove the polynomial growth bound discussed above. Note added: after this work was completed, we learned about the recent preprint of Matthew Folz [4] which contains related material in the special case of locally finite graphs. In particular, the main result [4, Theorem 1.3] is a special case of Corollary 4.4 and [4, Theorem 1.4] gives slightly better estimates than Corollaries 4.4 and 4.6.

261 886 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI 2. Preliminaries In this section, we introduce the basic notions and concepts. The first subsection is devoted to recalling the setting of Dirichlet forms. In the second subsection, we prove an estimate for the bottom of the essential spectrum and in the third subsection we discuss the basic properties of the test functions that are used to prove our result Dirichlet forms In this section, we recall some elementary facts about Dirichlet forms; see, for example, [7] and, for recent work on non-local forms [5]. As above, let X be a locally compact separable metric space and let m be a positive Radon measure of full support. We consider all functions on X to be real valued, but, by complexifying the corresponding Hilbert spaces and forms, we could also consider complex-valued functions. A closed non-negative form on L 2 (X, m) consists of a dense subspace D L 2 (X, m) and a sesquilinear non-negative map E : D D R such that D is complete with respect to the form norm E = E(, )+ 2, where always denotes the L 2 norm. We write E(u) :=E(u, u) for u D. A closed non-negative form (E,D) is called a Dirichlet form if, for any u D and any normal contraction c : R R, we have c u D and E(c u) E(u). Here, c is a normal contraction if c(0) = 0 and c(x) c(y) x y for x, y R. A Dirichlet form is called regular if D C c (X) is dense both in (D, E ) and (C c (X), ), where C c (X) is the space of continuous compactly supported functions. A function f : X R is said to be quasi-continuous if, for every ε>0, there is an open set U X with cap(u) :=inf{ v E v D, 1 U v} ε, such that f X\U is continuous (where inf = and 1 U is the characteristic function of U). For a regular Dirichlet form (E, D), every u D admits a quasi-continuous representative; see [7, Theorem 2.1.3]. In the following, we assume that, when considering u as a function, we always choose a quasi-continuous representative. There is a fundamental representation theorem for regular Dirichlet forms called the Beurling Deny formula; see [7, Theorems and 5.2.1]. It states that E(u) = dγ (c) (u)+ (u(x) u(y)) 2 dj(x, y)+ u(x) 2 dk(x), X X X\d where we choose a quasi-continuous representative of u in the second and third integrals. Here, Γ (c) is a positive semidefinite bilinear form on D D, with values in the signed Radon measures on X, whichisstrongly local, that is, satisfies Γ (c) (u, v) =0ifu is constant on the support of v. Secondly, J is a non-negative Radon measure on X X \ d (which is X X without the diagonal d := {(x, x) x X}) and, finally, k is a non-negative Radon measure on X. The first term on the right-hand side is called the strongly local part of E, the second term is called the jump part and the third term is called the killing term. The measure J gives rise to a form Γ (j) on D D with values in the signed Radon measures on X (where the j refers to jump ) which is characterized by dγ (j) (u) = (u(x) u(y)) 2 dj(x, y), K K X\d for K X compact and u D. The focus of this paper is on regular Dirichlet forms E without killing term, that is, k 0. Thus, we define Γ:=Γ (c) +Γ (j). X

262 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 887 The space Dloc of functions locally in the domain of E was introduced in [5] and is important for the definition of intrinsic metrics. It is defined as the set of functions u L 2 loc (X, m) such that for all open and relatively compact sets G there is a function v D such that u and v agree on G and for all compact K X, (u(x) u(y)) 2 dj(x, y) <. K X\d We can extend Γ (c) and Γ (j) to Dloc ; see [7, Remarks after the proof of Theorem 3.2.2] and [5, Proposition 3.3]. For the strongly local part, we have a chain rule (see [7, Theorem 3.2.2]) as follows: for ϕ : R R continuously differentiable with bounded derivative ϕ, Γ (c) (ϕ(u),v)=ϕ (u)γ (c) (u, v), u,v Dloc L (X, m). A pseudometric is a map ρ : X X [0, ] which is symmetric, satisfies the triangle inequality and ρ(x, x) = 0 for all x X. ForA X, we define the map ρ A : X [0, ] by ρ A (x) = inf ρ(x, y). y A If ρ is a pseudometric and T>0, then ρ T is a pseudometric and we have (ρ T ) A = ρ A T and ρ A (x) T ρ A (y) T ρ(x, y). By Frank, Lenz and Wingert [5, Definition 4.1], a pseudometric ρ is called an intrinsic metric for the Dirichlet form E if there are Radon measures m (c) and m (j) with m (c) + m (j) m such that for all A X and all T>0the functions ρ A T are in Dloc C(X), where C(X) denotes the set of continuous functions on X, and satisfy Γ (c) (ρ A T ) m (c) and Γ (j) (ρ A T ) m (j). This implies that if A X is such that ρ A (x) < for all x X, then ρ A Dloc C(X) and Γ(ρ A ) m (see [5, Proposition 4.4]) An estimate for the bottom of the essential spectrum The following Persson-type theorem seems to be standard in certain settings; see [10, 26]. However, since we are not able to find a proper reference in the literature which covers our case, we include a short proof. Proposition 2.1. Let h be a closed quadratic form on L 2 (X, m) that is bounded from below and let H be the corresponding self-adjoint operator. Assume that there is a normalized sequence (f n ) in D(h) that converges weakly to zero in L 2 (X, m). Then, λ ess 0 (H) lim inf h(f n). n Proof. Without loss of generality, assume h 0andλ ess 0 (H) > 0. Let 0 <λ<λ ess 0 (H). We will show that there is an N 0 such that h(f n ) >λfor all n N. Letλ 1 be such that λ<λ 1 <λ ess 0 (H) and let ε>0bearbitrary. As λ 1 <λ ess 0 (H), the spectral projection E (,λ1 ] of H and the interval (,λ 1 ] is a finite rank operator. Therefore, as (f n ) converges weakly to zero, there is an N 0 such that E (,λ1 ]f n 2 <εfor n N. Letting ν n be the spectral measure of H with respect to f n, we estimate for n N, h(f n ) tdν n (t) λ 1 dν n (t) =λ 1 ( f n 2 E (,λ1 ]f n 2 ) >λ 1 (1 ε), λ 1 λ 1 where we used λ 1 0ash 0. We conclude the asserted inequality by choosing ε =(λ 1 λ)/λ 1 > 0.

263 888 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI 2.3. The test functions In this section, we introduce the sequence of test functions that we will use to estimate the bottom of the (essential) spectrum. For r N,x 0 X, α > 0, define f r,x0,α : X [0, ), x ((e αr e α(2r ρ(x 0,x)) ) 1) 0. Then, for fixed r, α, x 0,wehavef Br e αr 1, f B2r \B r = e α(2r ρ(x 0, )) 1andf X\B2r 0. Clearly, f is spherically homogeneous, that is, there exists h :[0, ) [0, ) such that f(x) = h(ρ(x 0,x)). The definition of f combines ideas from [2, 6, 28]. Moreover, for r N, x 0 X, α>0, let g r,x0,α : X [0, ) be given by g r,x0,α =(f r,x0,α + 2)1 B2r. Lemma 2.2. Let x 0 X, f r = f r,x0,α and g r = g r,x0,α for r 0, α>0. Ifμ<, then we have the following conditions: (a) f r,g r L 2 (X, m) for all r 0, α>0; (b) if m( r B r)=, then f r / f r converges weakly to 0 as r ; (c) if α>μ/2, then there is a sequence (r k ) such that g rk / f rk 1 as k. In general (possibly μ = ), (d) if α> μ/2, then there are sequences (x k ) in X and (r k ) such that f k = f rk,x k,α, g k = g rk,x k,α L 2 (X, m) andwehavethat g k / f k 1 as k. Proof. (a) Since we assume μ<, it follows that m(b r (x 0 )) < for all r 0. Therefore, f r,g r L 2 (X, m) for all r 0 since f r and g r are bounded and supported in B 2r. (b) Let ψ L 2 (X, m) with ψ =1,ε>0 and set ϕ = ψ1 B r. There exists R>0 such that ϕ1 X\BR ε/2. Moreover, let r R be such that m(b R ) ε 2 m(b r )/4 (this choice is possible since m( B r )= ). We conclude by the Cauchy Schwarz inequality and f r 1 BR ε/2 f r that ϕ, f r = ϕ1 BR,f r + ϕ1 X\BR,f r ϕ f r 1 BR + ϕ1 X\BR f r ε f r. As supp f r s B s, it follows that ψ, f r = ϕ, f r for r 0 which proves (b). Before we prove (c), we show (d) and indicate how to adapt the proof to (c) afterwards. If μ =, then there is nothing to prove so assume μ <. Let 0 <ε<α μ/2. By the definition of μ, there are sequences (r k ) of increasing positive numbers and (x k ) of elements in X such that m(b 2rk (x k )) e (2 μ+ε)r k, k 0. m(b 1 (x k )) We set f k = f rk,x k,α, g k = g rk,x k,α. Since m(b 2rk (x k )) < and the functions f k,g k are supported in B 2rk (x k ) and bounded, they are in L 2 (X, m). By definition g k = g k 1 B2rk = (f k + 2)1 B2rk, k 0. Using the inequalities (a + b) 2 (1/(1 ε))a 2 +(1/ε)b 2 and f k 2 m(b rk (x k ))(e αr k 1) 2 m(b rk (x k )) e 2αr k /c for c =(1 e αr 0 ) 2 > 0, we get g k 2 f k 2 ( f k +2 m(b2rk (x k ))) 2 f k 2 (1/(1 ε)) f k 2 +(4/ε)m(B 2rk (x k )) f k 2 1 (1 ε) + 4c m(b 2rk (x k )) ε m(b rk (x k )) e 2αr k.

264 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 889 For r k large enough, we have by definition of μ, m(b rk (x k )) m(b 1 (x k )) m(b rk (x)) inf x X m(b 1 (x)) e( μ ε)r k. Thus, by the choice of (r k ) and (x k ), we have m(b 2rk )/m(b rk ) e ( μ+2ε)r k.as0<ε<α μ/2, g k 2 f k 2 1 (1 ε) + 4c e( μ+2ε 2α)r k 1 as k. ε (1 ε) Since ε can be chosen arbitrarily small and g k f k, we deduce the statement. For (c), we choose (x k )tobex 0 and follow the lines of the proof replacing μ by μ. Remark 2.3. as k. If inf x X m(b 1 (x)) > 0, then f k / f k of (d) also converges weakly to zero The following auxiliary estimates will later give us bounds for the Lipshitz constants of f r,x,α. Lemma 2.4. Let α>0. For all R 0, one has (e αr 1) 2 (e 2αR +1) α2 R 2. 2 Moreover, for R [0, 1] one has (e αr 1) 2 (e 2αR +1) R2 (e α 1) 2 (R 2 e 2α +1). Proof. For the first statement, let s = αr and check via a series expansion that s s 2 (e 2s +1) 2(e s 1) 2 is non-negative. The second statement follows by e αr 1 R(e α 1) for R [0, 1] from the series expansion, the elementary inequality a 2 /((a +1) 2 + 1) b 2 /((b +1) 2 +1)for0 a b, and 1 R 0. Lemma 2.5. Let r N, x 0 X, α > 0 and set f := f r,x0,α, g:= g r,x0,α. Then, for x, y X, (f(x) f(y)) 2 c(α)(g(x) 2 + g(y) 2 )ρ(x, y) 2, where c(α) =α 2 /2. If, additionally, ρ(x, y) 1, then c(α) canbechosentobec(α, ρ(x, y)) = (e α 1) 2 /ρ(x, y) 2 e 2α +1. In particular, f is Lipshitz continuous with Lipshitz constant α(e αr +1). Proof. We fix r, α and x 0 for the proof. Let x, y X be given and let s = ρ(x 0,x)andt = ρ(x 0,y). We define D s,t := (f(x) f(y)) 2. Moreover, we estimate F (R) := (e αr 1) 2 /e 2αR +1, R 0byc(α)R 2 (and by c(α, R)R 2 for R 1), from Lemma 2.4. By symmetry, we may assume, without loss of generality, that s t so that we have six cases to check. Case 1: If s t r, then D s,t =0. Case 2: If s r t 2r, then since t r t s = ρ(x 0,y) ρ(x 0,x) ρ(x, y) andg(x) = e αr +1,g(y) =e α(2r t) +1, D s,t =(e αr e α(2r t) ) 2 =(e 2αr + e 2α(2r t) )F (t r) (e 2αr + e 2α(2r t) )c(α)(t r) 2 c(α)(g(x) 2 + g(y) 2 )ρ(x, y) 2.

265 890 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI Case 3: If s r 2r t, then since r t s ρ(x, y), g(x) =e αr +1andg(y) =0, D s,t =(e αr 1) 2 =(e 2αr +1)F (r) (e 2αr +1)c(α)r 2 c(α)(g(x) 2 + g(y) 2 )ρ(x, y) 2. Case 4: If r s t 2r, then since t s ρ(x, y)andg(x) =e α(2r s) +1,g(y) =e α(2r t) +1, D s,t =(e α(2r s) e α(2r t) ) 2 =(e 2α(2r s) + e 2α(2r t) )F (t s) c(α)(g(x) 2 + g(y) 2 )ρ(x, y) 2. Case 5: If r s 2r t, then since 2r s t s ρ(x, y), g(x) =e α(2r s) +1andg(y) =0, D s,t =(e α(2r s) 1) 2 =(e 2α(2r s) +1)F (2r s) c(α)(g(x) 2 + g(y) 2 )ρ(x, y) 2. Case 6: If 2r s t, then D s,t =0. The Lipshitz bound follows since g is bounded by e αr +1. Lemma 2.6. Let (E,D) be a regular Dirichlet form and let ρ be an intrinsic metric. For all r>0, x 0 X and α>0, we have f r,x0,α Dloc. Moreover, if B 2r(x 0 ) is compact, then f r,x0,α D. Proof. By Lemma 2.5, the functions f := f r,x0,α are Lipshitz continuous for all r>0, x 0 and α>0. Thus, by a Rademacher-type theorem (see [5, Theorem 4.8] or, for strongly local forms, see [27, Theorem 5.1]), we have f D loc and Γ(f) m. IfB 2r(x 0 ) is compact, then f is compactly supported which implies f D (see [5, Theorem 3.4]) The strongly local estimate 3. Proof of the main theorem In this subsection, we give an estimate which will be used to prove the theorem for the strongly local part of the Dirichlet form. For given r N, x 0 X and α>0, we define f := f r,x0,α and g := g r,x0,α. Lemma 3.1. Let ρ be an intrinsic metric for a regular strongly local Dirichlet form E. Then, for all r>0, x 0 X and α>0 such that f D, wehave E(f) α 2 g 2 dm (c). X Proof. get that As E is strongly local, by the chain rule and the fact that ρ is an intrinsic metric we E(f) = dγ (c) (f) = dγ (c) (e α(2r ρ(x0, )) 1) B 2r \B r B 2r \B r = α B 2 e 2α(2r ρ(x0, )) dγ (c) (ρ(x 0, )) 2r \B r α 2 e 2α(2r ρ(x0, )) dm (c) α 2 gr,x 2 0,α dm (c). B 2r \B r X 3.2. The non-local estimate Next, we treat the non-local case. With applications to graphs in the next section in mind, we do not assume that the jump part is a regular Dirichlet form for now.

266 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 891 For this subsection, let m be a Radon measure on X and let J be a symmetric Radon measure on X X \ d such that for every m-measurable A X, the set A X \ d is J measurable and vice versa. Let ρ be a pseudometric on X which is J measurable and assume that for all measurable A X, ρ(x, y) 2 dj(x, y) m(a), ( ) A X\d which immediately implies that for all measurable functions ϕ, ϕ(x) 2 ρ(x, y) 2 dj(x, y) ϕ 2 dm. X X\d We say that the pseudometric ρ has jump size in [a, b], 0 a b, if for the set A a,b := {(x, y) X X ρ(x, y) [a, b]}\d, ϕ(x, y) dj(x, y) = ϕ(x, y) dj(x, y), X X\d A a,b for any positive measurable function ϕ. For given r N, x 0 X and α>0, we define f := f r,x0,α and g := g r,x0,α. X Lemma 3.2. Assume that ρ satisfies ( ). For all r N, x 0 X and α>0, (f(x) f(y)) 2 dj(x, y) 2c(α) g 2 dm, X X\d X where c(α) =α 2 /2. Ifρ has jump size in [δ, 1] for some 0 δ 1, then c(α) can be chosen to be c(α, δ) =(e α 1) 2 /1+δ 2 e 2α. Proof. By Lemma 2.5 and since ρ satisfies ( ), (f(x) f(y)) 2 dj(x, y) α 2 g(x) 2 ρ(x, y) 2 dj(x, y) α 2 X X\d X X\d X g 2 dm. Let δ>0. If ρ has jump size in [δ, 1], then (f(x) f(y)) 2 dj(x, y) = (f(x) f(y)) 2 dj(x, y) X X\d A δ,1 g(x) Aδ,1 2 2(e α 1) 2 (1 + ρ(x, y) 2 e 2α ) ρ(x, y)2 dj(x, y) 2(eα 1) 2 (1 + δ 2 e 2α g(x) 2 ρ(x, y) 2 dj(x, y) ) X X\d 2(eα 1) 2 (1 + δ 2 e 2α g 2 dm. ) 3.3. Proof of Theorem 1.1 We now have all of the ingredients to prove our main result. ProofofTheorem1.1. By Frank, Lenz and Wingert [5, Lemma 4.7], an intrinsic metric satisfies ( ). Moreover, under the assumption that the distance balls are compact we have f r,x,α D for all r>0, x X, α>0 by Lemma 2.6. X

267 892 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI Let α> μ/2. By Lemma 2.2(d), there are sequences (x k ) and (r k ) such that for f k = f rk,x k,α, g k = g rk,x k,α, λ 0 (L) lim inf k E(f k ) f k 2 α2 lim k g k 2 f k 2 = α2, where the second inequality follows from Lemmas 3.1 and 3.2 and the equality follows from Lemma 2.2(d). Hence, λ 0 (L) μ 2 /4. Let now α>μ/2andlet(r k ) be the sequence given by Lemma 2.2(c) for some fixed x 0 X and let x k = x 0 for all k 0. By Lemma 2.2(b), the sequence (f k / f k ) converges weakly to zero in L 2 (X, m) and, therefore, by Proposition 2.1, Lemmas 3.1 and 3.2, and Lemma 2.2(c) we get λ ess E(f k ) 0 (L) lim inf α2 lim k f k 2 Therefore, λ ess 0 (L) μ 2 / A more general non-local estimate k g k 2 f k 2 = α2. Assume that a closed, symmetric, non-negative form (E J,D J ) is given by E J (f) = X X\d (f(x) f(y))2 dj(x, y) forameasurej such as introduced in Subsection 3.2. Let L be the positive self-adjoint operator associated to (E J,D J ). Theorem 3.3. Assume that ρ satisfies ( ) and f r,x,α D J for all r 0, x X and α> μ/2. Then, λ 0 (L) μ2 4 and if m( B r (x 0 )) = for x 0 used to define μ. If the jump size is in [δ, 1] for some 0 δ 1, then 2(e μ/2 1) 2 λ 0 (L) δ 2 e μ and +1 if m( B r (x 0 )) = for x 0 used to define μ. λ ess 0 (L) μ2 4 λ ess 0 (L) 2(eμ/2 1) 2 δ 2 e μ +1 Proof. The proof follows along the same lines as the proof of the main theorem, Theorem 1.1, by using Proposition 2.1, Lemmas 2.2 and Weighted graphs 4. Applications In this section, we derive consequences of Theorems 1.1 and 3.3 for graphs. We briefly introduce the setting and refer for more background to [18]. Let X be a countable discrete set. Every Radon measure of full support on X is given by a function m : X (0, ). Then, L 2 (X, m) is the space l 2 (X, m) ofm-square summable functions with norm u =( x u(x)2 m(x)) 1/2, u l 2 (X, m). From [18, Theorem 7], it can be seen that all regular Dirichlet forms on (X, m) without killing term are determined by a symmetric map b : X X [0, ) with vanishing diagonal that satisfies b(x, y) < for all x X, y X

268 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 893 and gives rise to a measure J =1/2b on X X \ d (as measures on discrete spaces are determined by non-negative functions). The 1 2 stems from the convention that we consider each edge only once in the form. The map b can then be interpreted as a weighted graph with vertex set X. Namely, the vertices x, y X are connected by an edge with weight b(x, y) ifb(x, y) > 0. In this case, we write x y. A graph is called connected if, for all x, y X, there are vertices x i X, i =1,...,n such that x = x 0 x 1 x n = y. We call (x 0,x 1,...,x n )apath connecting x and y. We say the graph is locally finite if for each x X the number of y X such that x y is finite. Let a map Ẽ : l2 (X, m) [0, ] be given by Ẽ(u) = 1 b(x, y)(u(x) u(y)) 2. 2 x,y X The regular Dirichlet form E associated to J is the restriction of Ẽ to C c(x) E. Moreover, let E max = Ẽ D max, Dmax = {u l 2 (X, m) Ẽ(u) < }, which is also a Dirichlet form that is, in general, not regular. We denote the operator arising from E by L and the operator arising from E max by L max. Let ρ be an intrinsic metric on X. In the context of graphs, this is equivalent to ( ) (see [5, Lemma 4.7, Theorem 7.3]) which reads as 1 b(x, y)ρ(x, y) 2 m(x) for all x X. 2 y X For simplicity, we restrict ourselves to the case when ρ takes values in [0, ). (Otherwise, we can easily consider the graph componentwise.) Remark 4.1. Very often it is convenient to consider intrinsic metrics which satisfy y X b(x, y)ρ(x, y)2 m(x) for all x X (that is, we drop the 1 2 on the left-hand side). For example, in [14, 15] an explicit example of such a metric ρ is given, for x, y X, by ρ(x, y) :=inf{l(x 0,...,x n ) n 1, x 0 = x, x n = y, x i x i 1,i=1,...,n}, where the length l is given by l(x 0,...,x n )= n i=1 min{deg(x i) 1/2, Deg(x i 1 ) 1/2 } and Deg(z) = w b(z,w)/m(z) is a generalized vertex degree. In this case, all estimates in the theorem below can be divided by 2. In general, it is hard to determine whether the distance balls with respect to a certain metric are compact, meaning finite in the original topology. However, we always have a statement for the operator L max related to E max. Theorem 4.2. Assume that b is connected and m(x) =. Then, λ 0 (L max ) μ2 4 If ρ(x, y) [δ, 1] for all x y, then λ 0 (L max 2(e μ/2 1) 2 ) δ 2e μ +1 and and λ ess 0 (L max ) μ2 4. λ ess 0 (L max ) 2(eμ/2 1) 2 δ 2 e μ +1. Remark 4.3. If the assumption on the intrinsic metric in the theorem above is posed without the 1 2 on the left-hand side, then all estimates can be divided by 2.

269 894 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI Proof. Let B rk (x 0 ), (respectively, B rk (x k )) be a sequence of distance balls that realizes μ (respectively, μ), that is, μ = lim inf k r 1 k log m(b r k (x 0 )) (respectively, μ = lim inf k r 1 k log m(b r k (x k ))). If the measure of B rk (x 0 ) (respectively, B rk (x k )) is infinite for some k, then μ = (respectively, μ = ) and we are done. Otherwise, f rk,x 0,α,g rk,x 0,α l 2 (X, m) (respectively, f rk,x k,α, g rk,x k,α l 2 (X, m)) and f rk,x 0,α D max (respectively, f rk,x k,α D max ) by Lemma 3.2. Thus, the statement follows directly from Theorem 3.3. If we know more about either the measure or the metric structure, then we can say something about the operator L. This is the case under either of the following additional assumptions: (A) every infinite path of vertices has infinite measure; (B) ρ is an intrinsic path metric with bounded jump size on a locally finite graph such that (X, ρ) is metrically complete. In particular, (A) is satisfied if inf x X m(x) > 0 and (B) is satisfied if all infinite geodesics have infinite length. Corollary 4.4. Assume that either (A) or (B) is satisfied. Then, the statement of Theorem 4.2 holds for L = L max. Proof. By Keller and Lenz [18, Theorem 6], assumption (A) implies E = E max and L = L max. Moreover, by Huang, Keller, Masamune and Wojciechowski [16, Theorem 2], assumption (B) also implies E = E max and L = L max. Remark 4.5. Under the slightly stronger assumption that connected infinite sets have infinite measure we can prove the corollary directly. Namely, if one of the relevant distance balls is infinite, then it has infinite measure and the exponential volume growth is infinite. In the other case, the corollary follows from Theorem 3.3. We also recover the result of [6] which already covers [3, 25]. In their very particular situation, m is the vertex degree and b takes values in {0, 1}. The natural graph distance d is given as the minimum length of a path of edges connecting two vertices where the length is the number of edges contained in the path. Corollary 4.6 (Normalized Laplacians). Let b be a connected weighted graph over (X, n), with the measure n(x) = y X b(x, y), x X and let d be the natural graph metric. Then, λ 0 (L) 1 2e μ/2 /(1 + e μ ) and λ ess 0 (L) 1 2e μ/2 /(1 + e μ ). Proof. Clearly, L is a bounded operator and thus L = L max. Moreover, the natural graph metric is an intrinsic metric for 2L and its jump size in exactly 1. Thus, the statement follows from Theorem Standard graphs and the natural graph distance Let b : X X {0, 1} and m 1. We call such a graph standard. In this case, the operator L becomes the graph Laplacian Δ acting on D(Δ) = {ϕ l 2 (X) (x y x (ϕ(x) ϕ(y)))

270 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 895 l 2 (X)} (see [18, 30]) as Δϕ(x) = y x(ϕ(x) ϕ(y)), where x y means that b(x, y) =1.Bym 1, we have that m(a) = A for all A X. For simplicity, we assume that the graph is connected and infinite. Theorem 4.7. Let d be the natural graph distance and B d r (x 0 )={x X d(x, x 0 ) r} for some x 0 X and r 0. If lim inf r then λ 0 (Δ) = λ ess 0 (Δ) = 0. Moreover, if lim sup r log B d r (x 0 ) log r then λ ess 0 (Δ) < and, in particular, σ ess (Δ). < 3, B d r (x 0 ) r 3 <, Remark 4.8. (a) The result above is sharp. This can be seen by the examples of antitrees discussed after the proof. (b) In [9, Theorem 1.4], it is shown that less than cubic growth implies stochastic completeness. (c) If the vertex degree is bounded by K, then the situation is very different: the n in Corollary 4.6 becomes deg in this case, where deg : X N is the function assigning to a vertex the number of adjacent vertices, and the corresponding normalized operator Δ acts on l 2 (X, deg) as Δϕ(x) =(1/deg(x)) y x (ϕ(x) ϕ(y)). Then, λ 0 ( Δ) λ 0 (Δ) Kλ 0 ( Δ) and λ ess 0 ( Δ) λ ess 0 (Δ) Kλ ess 0 ( Δ); see, for example, [17]. Thus, in the case of bounded degree, the threshold again lies at subexponential growth by Corollary 4.6 (as the measures m 1andn = deg also give the same exponential volume growth). Explicit estimates for the exponential volume growth of planar tessellations in terms of curvature can be found in [20]. (d) In the case of bounded vertex degree, we also have a threshold for the recurrence of the corresponding random walk at quadratic volume growth; see [29, Lemma 3.12]. Let ρ be the intrinsic metric from [14] introduced above in Remark 4.1 which, in the case of standard graphs, is given by { n 1 } ρ(x, y) =inf min{deg(x i ) 1/2, deg(x i+1 ) 1/2 } (x 0,...,x n ) is a path from x to y. i=0 Let B ρ r = {x X ρ(x, x 0 ) r}, while B d r are balls with respect to the natural graph distance d for some fixed x 0 X. The proof of the theorem is based on the following lemma which is inspired by the proof of [9, Theorem 1.4]. Indeed, the second statement is taken directly from there. Lemma 4.9. If lim inf r log B d r /log r = β [1, 3), then lim inf r log B ρ r /log r (2β/3 β). Moreover, if lim sup r B d r /r 3 <, then lim sup r (1/r) log B ρ r <.

271 896 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI Proof. Let Sr d = Br d \ Br 1, d r 0 and for convenience set S r d = B r d = for r>0. Let 1 α<3andlet(r k ) be an increasing sequence such that log Br d k (x 0 ) /log r k <α for all k 0. Then, B d r k = r k r=0 S d r <r α k, for large k 0. For ε>0andk 0, set { A k := r [0,r k ] N 0 Sr d > α ε α (r +1)α 1}. We can estimate A k εr k via rk α > Br d k α ε α (r +1) α 1 α A k ε α (r +1) α 1 α Ak ε α r α 1 dr = A k α r A k r=0 0 ε α. Thus, { r [1,r k ] N 0 max i=0,1,2,3 Sd r i > α } (r +1)α 1 4εr εα k and { r [1,r k ] N 0 max i=0,1,2,3 Sd r i α } (r +1)α 1 (1 4ε)r εα k. As deg Sr 1 d Sr d Sr+1 d on Sr d,weget D k (1 4ε)r k, where { D k := r [1,r k ] N 0 deg 3α } ε α (r +1)α 1 on Sr 2 d Sr 1 d. Hence, for r D k and x Sr 2, d y Sr 1, d ρ(x, y) c(r +1) (α 1)/2 with c = ε α /3α. Since any path from x 0 to Sr d k contains such edges, we have, for any x Sr d k, ρ(x 0,x) c r k rk (r +1) (α 1)/2 c r (α 1)/2 c r (α 1)/2 dr C 0 r (3 α)/2 k, r D k r=4εr k +2 4εr k +2 with C 0 > 0forε>0 chosen sufficiently small and r k large. Let R k := C 0 r (3 α)/2 k and C := C 2α/(3 α) 0. Then, B ρ R k Br d k and, since Br d k = r k r=0 Sr d <rk α, we conclude B ρ R k Br d k <rk α CR 2α/(3 α) k. Thus, the first statement follows. The second statement is shown in the proof of [9, Theorem 1.4]. Proof of Theorem 4.7. In the case where the polynomial growth is strictly less than cubic we get by the lemma above that μ = 0 with respect to the intrinsic metric ρ, and in the case where it is less than cubic we still have μ<. Thus, the statement follows from Corollary 4.4, where (A) is clearly satisfied since m 1. Let us discuss the example of antitrees which show the sharpness of the result. They were first introduced in [32] and further studied in [1, 19]. Example An antitree is a spherically symmetric graph, where a vertex in the rth sphere is connected to all vertices in the (r + 1)th sphere for r 0, and there are no horizontal

272 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 897 edges. Thus, an antitree is characterized by a sequence (s r ) taking values in N which encodes the number of vertices in the sphere Sr d = Br d \ Br 1. d Stronger growth than cubic. In[19, Corollary 6.6], it is shown that if an antitree has more than cubic polynomial volume growth (that is, r 3+ɛ forɛ > 0), then λ 0 (Δ) > 0andσ ess (Δ) =. Indeed, with respect to the intrinsic metric ρ, these antitrees have finite diameter and thus μ = ; see [14, Example 4.3.2]. Cubic growth. If the distance spheres of an antitree satisfy Sr d =(r +1) 2, then Br d (r +1) 3. Moreover, the function ϕ which takes the value r 2 on vertices of the (r 1)th sphere, r 1, is a positive generalized super-solution for Δ to the value 2, that is, Δϕ 2ϕ. Thus, by a discrete Allegretto Piepenbrink theorem (see [31, Theorem 4.1] or [11, Theorem 3.1]) it follows that λ 0 (Δ) 2. By Theorem 4.7, it follows that 2 λ ess 0 (Δ) <. Weaker growth than cubic. In this case, Theorem 4.7 shows λ 0 (Δ) = λ ess 0 (Δ) = 0. Acknowledgements. The authors are grateful to Józef Dodziuk and Daniel Lenz for their continued support and for generously sharing their knowledge. References 1. J. Breuer and M. Keller, Spectral analysis of certain spherically homogeneous graphs, Oper. Matrices, to appear. 2. R. Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981) J. Dodziuk and L. Karp, Spectral and function theory for combinatorial Laplacians, Geometry of random motion (Ithaca, NY, 1987), Contemporary Mathematics 73 (American Mathematical Society, Providence, RI, 1988) M. Folz, Volume growth and spectrum for general graph Laplacians, Math. Z. (2012) DOI: /s y 5. R. L. Frank, D. Lenz and D. Wingert, Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory, J. Funct. Anal., toappear. 6. K. Fujiwara, Growth and the spectrum of the Laplacian of an infinite graph, Tohoku Math. J. (2) 48 (1996) M. Fukushima, Y. Ōshima and M. Takeda, Dirichlet forms and symmetric Markov processes, degruyter Studies in Mathematics 19 (Walter de Gruyter, Berlin, 1994). 8. A. Georgakopoulos, S. Haeseler, M. Keller, D. Lenz and R. Wojciechowski. Graphs of finite measure, Preprint. 9. A. Grigor yan, X. Huang and J. Masamune, On stochastic completeness of jump processes, Math. Z. 271 (2012) G. Grillo, On Persson s theorem in local Dirichlet spaces, Z. Anal. Anwend. 17 (1998) S. Haeseler and M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, Random walks, boundaries and spectra, Progress in Probability 64 (Birkhäuser, Basel, 2011) Y. Higuchi, A remark on exponential growth and the spectrum of the Laplacian, Kodai Math. J. 24 (2001) Y. Higuchi, Boundary area growth and the spectrum of discrete Laplacian, Ann. Global Anal. Geom. 24 (2003) X. Huang, On stochastic completeness of weighted graphs, PhD Thesis, X. Huang, On uniqueness class for a heat equation on graphs, J. Math. Anal. Appl. 393 (2012) X. Huang, M. Keller, J. Masamune and R. K. Wojciechowski, A note on self-adjoint extensions of the Laplacian on weighted graphs, J. Funct. Anal. 265 (2013) M. Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346 (2010) M. Keller and D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, J. reine angew. Math. 666 (2012) M. Keller, D. Lenz and R. K. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Math. Z. 274 (2013) M. Keller and N. Peyerimhoff, Cheeger constants, growth and spectrum of locally tessellating planar graphs, Math. Z. 268 (2011) D. Lenz, P. Stollmann and I. Veselić, The Allegretto Piepenbrink theorem for strongly local Dirichlet forms, Doc. Math. 14 (2009) P. Li and J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001) P. Li and J. Wang, Counting cusps on complete manifolds of finite volume, Math. Res. Lett. 17 (2010)

273 898 VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 24. L. Notarantonio, Growth and spectrum of diffusions, Preprint, 1998, arxiv:math/ v Y. Ohno and H. Urakawa, On the first eigenvalue of the combinatorial Laplacian for a graph, Interdiscip. Inform. Sci. 1 (1994) A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand. 8 (1960) P. Stollmann, A dual characterization of length spaces with applications to Dirichlet metric spaces, Studia Math. 198 (2010) K.-T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p -Liouville properties, J. reine angew. Math. 456 (1994) W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138 (Cambridge University Press, Cambridge, 2000). 30. R. K. Wojciechowski, Stochastic completeness of graphs (ProQuest LLC, Ann Arbor, MI, 2008), Thesis (PhD), City University of New York. 31. R. K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J. 58 (2009) R. K. Wojciechowski, Stochastically incomplete manifolds and graphs, Random walks, boundaries and spectra, Progress in Probability 64 (Birkhäuser, Basel, 2011) Sebastian Haeseler and Matthias Keller Mathematisches Institut Friedrich Schiller Universität Jena D Jena Germany sebastian haeseler@uni-jena de mkeller@ma huji ac il Rados law K. Wojciechowski York College of the City University of New York Jamaica, NY USA rwojciechowski@gc cuny edu

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275 CHAPTER 12 F. Bauer, B. Hua, M. Keller, On the l p spectrum of Laplacians on graphs, Advances in Mathematics 248 (2013),

276 Available online at ScienceDirect Advances in Mathematics 248 (2013) On the l p spectrum of Laplacians on graphs Frank Bauer a,b,, Bobo Hua b, Matthias Keller c a Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA b Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, Leipzig, Germany c Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Received 29 November 2012; accepted 2 May 2013 Communicated by Andreas Dress Abstract We study the p-independence of spectra of Laplace operators on graphs arising from regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth condition. Moreover, under a mild assumption on the measure we show a one-sided spectral inclusion without any further assumptions. We study applications to normalized Laplacians including symmetries of the spectrum and a characterization for positivity of the Cheeger constant. Furthermore, we consider Laplacians on planar tessellations for which we relate the spectral p-independence to assumptions on the curvature Elsevier Inc. All rights reserved. Keywords: l p -spectrum; Discrete Laplace operator; Regular Dirichlet forms; Cheeger constant; Planar tessellation 1. Introduction In [41,42] Simon conjectured that the spectrum of a Schrödinger operator acting on L p (R N ) is p-independent. Hempel and Voigt gave an affirmative answer in [29] for a large class of potentials. Later this result was generalized in various ways. Sturm [43] showed p-independence of the spectra for uniformly elliptic operators on a complete Riemannian manifold with uniform subexponential volume growth and a lower bound on the Ricci curvature.moreover,arendt [1] proved p-independence of the spectra of uniformly elliptic operators in R N with Dirichlet or Neumann * Corresponding author at: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA. addresses: fbauer@math.harvard.edu (F. Bauer), bobo.hua@mis.mpg.de (B. Hua), mkeller@ma.huji.ac.il (M. Keller) /$ see front matter 2013 Elsevier Inc. All rights reserved.

277 718 F. Bauer et al. / Advances in Mathematics 248 (2013) boundary conditions under the assumption of upper Gaussian estimates for the corresponding semigroups. While the proof strategies of Sturm and Arendt are rather similar to the one used by Hempel and Voigt, Davies [11] gave a simpler proof of the p-independence of the spectrum under the stronger assumption of polynomial volume growth and Gaussian upper bounds using the functional calculus developed in [10]. In recent works p-independence of spectral bounds are proven in the context of conservative Markov processes [40,45] and Feynman Kac semigroups [6,13,46]. See also [24] for Laplace operators on graphs with finite measure. In this paper, we prove p-independence of spectra for Laplace operators on graphs under the assumption of uniform subexponential volume growth. This question was brought up in [12, p. 378] by Davies. Our framework are regular Dirichlet forms on discrete sets as introduced in [39]. While our result is similar to the one of Sturm [43] for elliptic operators on manifolds, we do not need to assume any type of lower curvature bounds nor any type of bounded geometry. However, in various classical examples, such as Laplacians with standard weights, this disparity is resolved by the fact that uniform subexponential growth implies bounded geometry in some cases. In further contrast to [43], we do not assume any uniformity of the coefficients in the divergence part of the operator such as uniform ellipticity and, additionally, we allow for positive potentials (in general potentials bounded from below). We overcome the difficulties resulting from unbounded geometry, by the use of intrinsic metrics. While this concept is well established for strongly local Dirichlet forms [44] it was only recently introduced for general regular Dirichlet forms by Frank, Lenz and Wingert in [20]. Since then, this concept already proved to be very effective for the analysis on graphs, see [3,18, 19,25,28,33 35] where it also appears under the name adapted metrics. Moreover, we employ rather weak heat kernel estimates (with the log term instead of a square) by Folz [18], which is a generalization of [9] by Davies. These weak estimates turn out to be sufficient to prove the p-independence. Of course, the condition on uniform subexponential growth is always expressed with respect to an intrinsic metric. Another result of this paper is the inclusion of the l 2 -spectrum in the l p -spectrum under the assumption of lower bounds on the measure only. As applications we discuss the normalized Laplace operator, for which we prove several basic properties of the l p spectra such as certain symmetries of the spectrum. Moreover, we discuss consequences of p-independence on the Cheeger constant and give an example of p-independence and superexponential volume growth. Finally, we consider the case of planar tessellations which relates curvature bounds to the volume growth. In particular, we use such curvature conditions to recover results of Sturm [43] in the setting of planar tessellations. The paper is organized as follows. In the next section we introduce the set up and present the main results. In Section 3 we show several auxiliary results in order to prove the main results in Sections 4 and 5. Applications to normalized Laplacians are considered in Section 6. The final section, Section 7, is devoted to planar tessellations and consequences of curvature bounds on the volume growth and p-independence. 2. Set up and main results 2.1. Graphs Assume that X is a countable set equipped with the discrete topology. A strictly positive function m : X (0, ) gives a Radon measure on X of full support via m(a) = x A m(x) for A X, so that (X, m) becomes a discrete measure space.

278 F. Bauer et al. / Advances in Mathematics 248 (2013) A graph over (X, m) is a pair (b, c). Here, c : X [0, ) and b : X X [0, ) is a symmetric function with zero diagonal that satisfies b(x,y) <, for x X. y X We say x and y are neighbors or connected by an edge if b(x,y) > 0 and we write x y. For convenience we assume that there are no isolated vertices, i.e., every vertex has a neighbor. We call b locally finite if each vertex has only finitely many neighbors. The function c can be interpreted either as one-way-edges to infinity or a potential or a killing term. The normalizing measure n : X (0, ) given by n(x) = y X b(x,y), for x X, often plays a distinguished role. In the case where b : X X {0, 1}, n(x) gives the number of neighbors of a vertex x. Ifn/m M for some fixed M>0, then we say the graph has bounded geometry Intrinsic metrics and uniform subexponential growth By a pseudo metric we understand a function d : X X [0, ) that is symmetric, has zero diagonal and satisfies the triangle inequality. Following [20], we call a pseudo metric d an intrinsic metric for a graph b on (X, m) if b(x,y)d(x,y) 2 m(x), for all x X. y X For example one can always choose the path metric induced by the edge weights w(x,y) = ((m/n)(x) (m/n)(y)) 1 2,forx y, cf. e.g. [33]. Moreover, we call s := sup { d(x,y) x,y X, x y } the jump size of d. Note that the natural graph metric d n (i.e., the path metric with weights w(x,y) = 1forx y) is intrinsic if and only if m n. However, in the case of bounded geometry, i.e., n/m M for some fixed M>0, the metric d n / M (which is equivalent to d n )isan intrinsic metric. Throughout the paper we assume that d is an intrinsic metric with finite jump size. For the remainder of the paper, we refer to the quintuple (X,b,c,m,d) whenever we speak of the graph. We denote the distance balls centered at a vertex x X with radius r 0byB r (x) := {y X d(x,y) r}. Similar to [43], we say the graph has uniform subexponential growth if for all ε>0 there is C ε > 0 such that m ( B r (x) ) C ε e εr m(x), for all x X, r 0. In Section 3.1 we discuss some implications of this assumption Dirichlet forms and graph Laplacians Denote by C c (X) the space of complex valued functions on X with compact support. The l p -spaces are given by

279 720 F. Bauer et al. / Advances in Mathematics 248 (2013) where l p := l p (X, m) := { f : X C f p < }, f := sup f(x), and f p := x X p [1, ], ( f(x) 1 m(x)) p p, p [1, ). x X Note that l (X, m) does not depend on m. For p [1, ], let the Hölder conjugate be denoted by p, that is 1 p + 1 p = 1. We denote the dual pairing of f l p (X, m), g l p (X, m) by f,g := x X f(x)g(x)m(x), which becomes a scalar product for p = 2. We define the sesqui-linear form Q with domain D(Q) l 2 by Q(f, g) = 1 2 b(x,y)( f(x) f(y))( g(x) g(y)) + c(x)f(x)g(x), x,y X x X D(Q) = C c (X) Q, where Q = (Q( ) ) 1 2 and Q(f ) = Q(f, f ). The form Q is a regular Dirichlet form on l 2 (X, m), see[23,39] and for complexification of the forms, see [27, Appendix B]. The corresponding positive selfadjoint operator L = L 2 on l 2 (X, m) acts as Lf (x) = 1 b(x,y) ( f(x) f(y) ) + c(x) m(x) m(x) f(x). y X Let L be the extension of L to { F = f : X C b(x,y) f(y) } < for all x X. y X We have C c (X) D(L) if (and only if) LC c (X) l 2 (X, m),see[39, Theorem 6]. In particular, this can easily seen to be the case if the graph is locally finite or if inf x X m(x) > 0. If m = n and c 0, then L is referred to as the normalized Laplacian. Moreover, L = L 2 gives rise to the resolvents G α = (L α) 1, α<0, and the semigroups T t = e tl, t 0. These operators, αg α and T t, are positivity preserving and contractive (as Q is a Dirichlet form), and therefore extend consistently to operators on l p (X, m), p [1, ] (either by monotone convergence or by density of l 2 l p in l p, p [1, ) and taking the dual operator on l 1 to get the operator on l ). The semigroups are strongly continuous for p<. See [8, Theorem 1.4.1] for a proof of these facts. We denote the positive generators of G α and T t on l p by L p, p [1, ), and the dual operator L 1 of L 1 on l by L (which normally does not have dense domain in l ). By [39, Theorem 9] we have that L p, p [1, ], are restrictions of L. Moreover, L p are bounded operators with norm bound 2C if (n + c)/m C, see e.g. [38, Theorem 11] or [27, Theorem 9.3]. Hence, bounded geometry is equivalent to boundedness of the operators for c 0. We denote the spectrum of the operator L p by σ(l p ) and the resolvent set by ρ(l p ) = C \ σ(l p ), p [1, ]. By duality σ(l p ) = σ(l p ), p [1, ]. Throughout this paper C always denotes a constant that might change from line to line.

280 F. Bauer et al. / Advances in Mathematics 248 (2013) Main results In this section, we state the main theorems of this paper. The first result is a discrete version of Sturm s theorem [43], whose proof is given in Section 4. Theorem 2.1. Assume the graph has uniform subexponential growth with respect to an intrinsic metric with finite jump size. Then, for any p [1, ], Remark. σ(l p ) = σ(l 2 ). (a) Other than [43], we do not assume any type of bounded geometry or any types of lower bounds on the curvature. For a discussion and examples see Section 3.1. (b) Sometimes loops in the graph are modeled by non-vanishing diagonal of b. However, the assumption that b has zero diagonal has no influence on our main results above as possible non-vanishing diagonal terms do not enter the operators. Such loops only have an effect on n. Thus, when b had non-vanishing diagonal, then one only had to be careful whenever one chooses m = n. Clearly, we can also allow for potentials c such that c/m is only bounded from below (as adding a positive constant shifts the l p spectra of the operators simultaneously). The following theorem shows that under an assumption on the measure one spectral inclusion holds without any volume growth assumptions. Theorem 2.2. If m is such that inf x X m(x) > 0, then, for any p [1, ], σ(l 2 ) σ(l p ). The proof of Theorem 2.2 is given in Section Preliminaries In this section we collect some results and facts that will be used for the proof of Theorem 2.1. Moreover, in the first subsection we discuss the relation of uniform subexponential growth and bounded geometry Consequences of uniform subexponential growth Lemma 3.1. Assume the graph has uniform subexponential growth. Then, for all ε>0 there is C>0 such that: (a) m(x) Ce εd(x,y) m(y), for all x,y X. (b) #B r (x) Ce εr, for all r 0, where #B r (x) denotes the number of vertices in B r (x). (c) y X e εd(x,y) C, for all x X.

281 722 F. Bauer et al. / Advances in Mathematics 248 (2013) Proof. To prove (a) let x,y X. Using the uniform subexponential growth assumption and x B d(x,y) (y) yields m(x) m(b d(x,y) (y)) Ce εd(x,y) m(y). Turning to (b), let x X and r 0. We obtain using (a) and the uniform subexponential growth assumption #B r (x) = y B r (x) m(y)/m(y) Ce εr m(b r(x)) m(x) C 2 e 2εr. The final statement (c) follows also by direct calculation using (b) (with ε 1 ) e εd(x,y) = y X r=1 y B r (x)\b r 1 (x) e εd(x,y) e ε(r 1) #B r (x) C e (ε1 ε)r. r=1 r=1 Hence, choosing ε 1 = ε/2 yields the statement. Remark. (a) Lemma 3.1 (b) implies finiteness of distance balls. On the other hand, finite jump size s implies that for each vertex x all neighbors of x are contained in B s (x). Hence, graphs with uniform subexponential growth and finite jump size are locally finite. (b) Finiteness of distance balls has strong consequences on the uniqueness of selfadjoint extensions. In particular, by [35, Corollary 1] implies that Q is the maximal form on l 2 and that the restriction of L 2 to C c (X) (whenever C c (X) D(L 2 )) is essentially selfadjoint. In the following we discuss examples to clarify the relation between uniform subexponential growth and bounded geometry in the discrete setting. Recall that we speak of bounded geometry if n/m is a bounded function which is a natural adaption to the situation of weighted graphs. In Example 3.2 below, we show that there are graphs with uniform subexponential growth and unbounded geometry. For completeness we also give an example of bounded geometry and exponential growth which is certainly well known. Example 3.2. (a) Uniform subexponential growth and unbounded geometry. LetX = N, m 1, c 0 and consider b such that b(x,y) = 0for x y 1, b(x,x + 1) = x for x 4N and b(x,x + 1) = 1 otherwise. Clearly, (n/m)(x) = n(x) = x + 1forx 4N and, thus, the graph has unbounded geometry. In particular, L p is unbounded for all p [1, ]. Moreover, let d be the path metric induced by the edge weights w(x,x + 1) = (n(x) n(x + 1)) 1 2.We obtain that d(x,y) ( x y 3)/(4 2 ) for all x,y X. Hence, m(b r (x)) = #B r (x) 2(8r + 6) which implies uniform subexponential growth. (b) Exponential growth and bounded geometry. Take a regular tree, c 0, set b to be one on the edges and zero otherwise and let m 1. This graph has bounded geometry but is clearly of exponential growth. It is apparent that the graph in Example 3.2 (a) above has bounded combinatorial vertex degree and the unbounded geometry is induced by the edge weights. So, one might wonder whether there are also examples with unbounded combinatorial vertex degree which is the criterion for unbounded geometry in the classical setting. The proposition below shows that this is impossible

282 F. Bauer et al. / Advances in Mathematics 248 (2013) under the assumptions of uniform subexponential growth and finite jump size. Recall that by the remark below Lemma 3.1 we already know that the graph must be locally finite. The combinatorial vertex degree deg is the function that assigns to each vertex the number of neighbors, that is deg(x) = #{y X b(x,y) > 0}, x X. Proposition 3.3. If the graph has uniform subexponential growth with respect to a metric with finite jump size s, then the combinatorial vertex degree is bounded. Proof. Suppose the graph has unbounded vertex degree, i.e., there is a sequence of vertices (x n ) such that deg(x n ) n 2 for all n 1. We show that there is a sequence of vertices z n such that m(b s (z n ))/m(z n ) is unbounded and thus the graph does not have uniform subexponential growth. If, for n 1, there is a neighbor y n of x n such that m(y n ) m(x n )/ deg(x n ), then we estimate using x n B s (y n ) m(b s (y n )) m(y n ) m(x n) m(y n ) deg(x n ) n. We set z n = y n in this case. If, on the other hand, m(y) m(x n )/ deg(x n ) for all neighbors y of x n, then m(b s (x n )) 1 m(x n ) m(x n ) deg(x m(x n ) n) deg(xn ) deg(x n ) n and set z n = x n in this case. Hence, we have proven the claim. Corollary 1. Assume there is D>0 such that b D and m 1/D. Then, uniform subexponential growth with respect to a metric with finite jump size implies bounded geometry. Proof. One simply observes that n/m D 2 deg and the statement follows from the proposition above. The lemma and the corollary above mean for the standard Laplacians p ϕ(x) = y x (ϕ(x) ϕ(y)) on lp (X, 1) and (n) p ϕ(x) = deg(x) 1 y x (ϕ(x) ϕ(y)) on lp (X, deg), that uniform subexponential growth implies bounded geometry, both in the sense of bounded n/m and also in the sense of bounded deg Lipschitz continuous functions We denote by Lip ε constant ε>0, i.e., the real valued bounded Lipschitz continuous functions with Lipschitz Lip ε := { ψ : X R ψ(x) ψ(y) εd(x, y), x, y X } l (X, m). Lemma 3.4. Let ε>0 and let s be the jump size of d. Then, for all ψ Lip ε : (a) e ψ is a bounded Lipschitz continuous function, in particular, e ψ D(Q) = D(Q). (b) 1 e ψ(x) ψ(y) εe εs d(x,y),forx y. (c) (e ψ(x) e ψ(y) )(e ψ(x) e ψ(y) ) 2ε 2 e εs d(x,y) 2,forx y.

283 724 F. Bauer et al. / Advances in Mathematics 248 (2013) Proof. The first statement of (a) follows from mean value theorem, that is for any x,y X we have e ψ(x) e ψ(y) ψ(x) ψ(y) e ψ εd(x, y)e ψ. Now, e ψ D(Q) D(Q) is a consequence of [25, Lemma 3.5]. The other inclusion follows since e ψ is also bounded and Lipschitz continuous. Similarly, we get (b) using the Taylor expansion of the exponential function 1 e ψ(x) ψ(y) = (ψ(x) ψ(y)) k k! k 1 εd(x,y) k 1 (εs) k 1 k! εd(x, y)e εs, and, similarly, using (e ψ(x) e ψ(y) )(e ψ(x) e ψ(y) ) =2 k 2N 3.3. Kernels (ψ(x) ψ(y)) k k! we get (c). Let A : D(A) l p l q, p,q [1, ] be a densely defined linear operator. We denote by A p,q the operator norm of A,i.e. A p,q = sup Af q. f D(A), f p =1 Note that any such operator A : D(A) l p l q, p<, with C c (X) D(A) admits a kernel k A : X X C such that Af (x) = x X k A (x, y)f (y)m(y) for all f D(A), x X, which can be obtained by k A (x, y) = 1 m(x)m(y) A1 y, 1 x, where 1 v (w) = 1ifw = v and 1 v (w) = 0 otherwise. We recall the following well-known lemma which shows that the operator norm of A : l p l q can be estimated by its integral kernel. Lemma 3.5. Let p [1, ) and let A be a densely defined linear operator with C c (X) D(A) l p. Then: (a) A p,q ( y k A(,y) p q m(y)) 1 p for q < and p (1, ), and A 1,q sup y k A (,y) q for q<. (b) A p, sup x k A (x, ) p and equality holds if p = 1. Proof. Statement (a) follows from the fact f p = sup g l p, g p =1 g,f and twofold application of Hölder inequality. The first part of (b) follows simply from Hölder inequality. For the second part note that A 1, sup x,y X A1 y (x) /m(y).

284 F. Bauer et al. / Advances in Mathematics 248 (2013) Heat kernel estimates We denote the kernel of the semigroup T t, t 0byp t. As the semigroups are consistent on l p, p [1, ], i.e., they agree on their common domains, the kernel p t does not depend on p. The following heat kernel estimate will be the key to p-independence of spectra of L p.itis proven in [18], based on [9], for locally finite graphs and c 0. However, on the one hand, local finiteness is not used in [18] for this result and, on the other hand, the remark below Lemma 3.1 shows that we are in the local finite situation anyway whenever we assume uniform subexponential growth. We conclude the statement for c 0 by a Feynman Kac formula. Lemma 3.6. We have for all t 0 and x,y X p t (x, y) ( m(x)m(y) ) 1 d(x,y) 2 d(x,y) log e 2et. Proof. Denote the semigroup of the graph (b, 0) by T t (0) and the kernel by p t (0) and correspondingly for (b, c) by T t and p t. For c 0 the estimate above is found in [18, Theorem 2.1] for p t (0). Now, by a Feynman Kac formula, see e.g. [14,26],wehave [ p t (x, y) = T t δ y (x) = E x e t 0 m c (Xs)ds δ y (X t ) ] [ E x δy (X t ) ] = T t (0) δ y (x) = p t (0) (x, y), where δ y = 1 y /m(y). This proves the claim. By basic calculus, we obtain the following heat kernel estimate. Lemma 3.7. For all β>0 there exists a constant C(β) such that for all t 0, x,y X p t (x, y) ( m(x)m(y) ) 1 2 e βd(x,y)+c(β)t. Proof. Let β>0 and r>0letf(r)= r log(r/2e) + βr. Direct calculation shows that the function f assumes its maximum on the domain (0, ) at the point r 0 = 2e β. In particular, setting C(β) = 2e β yields d(x,y) t log d(x,y) 2et β d(x,y) t + C(β), for all t>0 and x,y X. The statement follows now from the lemma above. 4. Proof for uniform subexponential growth In this section we prove Theorem 2.1 following the strategy of [43]. The proof is divided into several lemmas and as always we assume that d is an intrinsic metric with finite jump size s. Lemma 4.1. For every compact set K ρ(l 2 ) there is ε>0 and C< such that for all z K and all ψ Lip ε e ψ (L 2 z) 1 e ψ 2,2 C.

285 726 F. Bauer et al. / Advances in Mathematics 248 (2013) Proof. Let ε>0 and ψ Lip ε.bylemma 3.4 we have e ψ D(Q) = e ψ D(Q) = D(Q). Let Q ψ be the (not necessarily symmetric) form with domain D(Q ψ ) = D(Q) acting as Q ψ (f, g) := Q ( e ψ f,e ψ g ) Q(f, g). Application of Leibniz rule yields, for f D(Q ψ ), Qψ (f, f ) = 1 2 x,y X x,y X x,y X b(x,y) f(y) 2( e ψ(x) e ψ(y))( e ψ(x) e ψ(y)) b(x,y)f(y) ( f(x) f(y) )( 1 e ψ(y) ψ(x)) b(x,y)f(y) ( 1 e ψ(x) ψ(y))( f(x) f(y) ). Applying Cauchy Schwarz inequality, Lemma 3.4 (b) and (c) and the intrinsic metric property, gives ε 2 C x,y X ( f(x) 2 b(x,y)d(x,y) 2 + 2Cε Cε 2 f Cε f 2Q(f ) 1 2. x,y X ) 1 f(x) b(x,y)d(x,y) 2 Q(f ) 2 Hence, the basic inequality 2ab (1/δ)a 2 + δb 2 for δ>0 and a,b 0 (applied with a = Cε f 2 2 and b = Q(f ) 1 2 ) yields Q ψ (f, f ) ( Cε ) f δq(f ). δ This shows that Q ψ is Q bounded with bound 0. According to [36, Theorem VI.3.9] this implies that the form Q ψ + Q is closed and sectorial. It can be checked directly that the corresponding operator is e ψ L 2 e ψ with domain D ψ = e ψ D(L 2 ). Moreover, for K ρ(l 2 ) compact, we can choose ε, δ > 0 that 2 (C(1 + 1/δ)ε 2 e 2s + δl 2 ) (L 2 z) 1 2,2 < 1 for all z K since C is a universal constant. Therefore, again by [36, Theorem VI.3.9] this implies existence of C = C(K,ε) such that e ψ (L 2 z) 1 e ψ 2,2 = ( e ψ L 2 e ψ z ) 1 2,2 C for all z K and ψ Lip ε. Let us recall some well-known facts about consistency of semigroups and resolvents. By [8, Theorem 1.4.1] the semigroups T t are consistent on l p. By the spectral theorem the Laplace transform for the resolvent G z = (L 2 z) 1 G z f = 0 e zt T t fdt, holds for f in l 2 and z {w C Rw<0} (the open left half plane). By density and duality arguments, this formula extends to f in l p, p [1, ), in the strong sense and to p = in

286 F. Bauer et al. / Advances in Mathematics 248 (2013) the weak sense. This shows that the resolvents (L p z) 1 are consistent on l p for z {w C Rw<0}. We denote by g α the kernel of the resolvent G α = (L p α) 1 which is independent of p [1, ] for α<0. Lemma 4.2. Assume the graph has uniform subexponential volume growth. For any ε>0 there exists α<0 and C< such that (a) g α (x, y) C(m(x)m(y)) 1 2 e εd(x,y) for all x,y X, (b) e ψ G α e ψ m 1 2 1,2 C for all ψ Lip ε, (c) m 1 2 e ψ G α e ψ 2, C for all ψ Lip ε. Proof. (a) By the Laplace transform of the resolvent and Lemma 3.7, we get g α (x, y) = 0 e αt p t (x, y) dt ( m(x)m(y) ) 1 2 e εd(x,y) 0 e (α+c)t dt, which yields the statement for α< C. (b) By Lemma 3.5 (a), ψ Lip ε, part (a) above (with ε 1 2ε) and Lemma 3.1 (c) e ψ G α e ψ m sup g 1,2 α (,y)e ψ( ) ψ(y) m(y) y X C sup e (2ε 2ε1)d(x,y) <. y X x X The proof of (c) works similarly using Lemma 3.5 (b). Lemma 4.3. Assume the graph has uniform subexponential volume growth. Then, (L 2 z) 2 extends to a bounded operator on l p for all z ρ(l 2 ) and p [1, ]. Moreover, for all compact K ρ(l 2 ) there is C< such that for all z K and p [1, ] (L 2 z) 2 p,p C. Proof. For z ρ(l 2 ) denote by g (2) z the kernel of the squared resolvent (G z ) 2 = (L 2 z) 2. Applying the resolvent identity twice yields (G z ) 2 = ( G α + (z α)g α G z )( Gα + (z α)g z G α ) = Gα ( I + (z α)gz ) 2Gα, for all α<0. Therefore, m 1 2 e ψ (G z ) 2 e ψ m 1 2 = ( m 1 2 e ψ G α e ψ)( I + (z α)e ψ/2 G z e ψ/2) 2( e ψ G α e ψ m 1 2 ), for all ε>0 and ψ Lip ε. Taking the norm 1, and factorizing... 1, (...) 2, (...) 2,2 (...) 1,2 yields that U := m 1 2 e ψ G 2 z e ψ m 1 2 is a bounded operator l 1 l by Lemma 4.1 and Lemma 4.2 with appropriate choice of α<0, ε>0 and all ψ Lip ε. Hence, the operator U admits a kernel k U (x, y) = (m(x)m(y)) 1 2 e ψ(x) ψ(y) g z (2) (x, y), x,y X, and we conclude from Lemma 3.5 (b) that g z (2) (x, y) C ( m(x)m(y) ) 1 2 e ψ(y) ψ(x).

287 728 F. Bauer et al. / Advances in Mathematics 248 (2013) For chosen ε>0 and any fixed x,y X let ψ : u ε(d(u,y) d(x,y)) and we obtain from Lemma 3.1 (a) (with ε) g (2) z (x, y) C ( m(x)m(y) ) 1 2 e εd(x,y) Cm(x) 1 e ε 2 d(x,y). Thus, using Lemma 3.5 (a) and Lemma 3.1 (c), we obtain 1,1 sup g (2) (x, y) m(x) C sup e 2 ε d(x,y) <. G 2 z y X x X z y X x X As G 2 z is bounded for p = 1 and p = 2, it follows from the Riesz Thorin interpolation theorem that it is bounded for p [1, 2] and by duality for p [1, ]. Lemma 4.4. If σ(l p ) [0, ) for all p [1, ], then σ(l 2 ) σ(l p ). Proof. The operators (L p z) 1 and (L p z) 1 are consistent for z {w C Rw<0} ρ(l p ) = ρ(l p ) by the discussion above Lemma 4.2 for all p [1, ]. By the assumption σ(l p ) [0, ) the resolvent sets are connected which yields by [30, Corollary 1.4] that (L p z) 1 and (L q z) 1 are consistent for z ρ(l p ) ρ(l q ) for p,q [1, ]. Moreover, by the standard theory [12, Lemma 8.1.3] (L p z) 1 and (L p z) 1 are analytic on ρ(l p ) = ρ(l p ). By the Riesz Thorin theorem these resolvents can be consistently extended to analytic l 2 -bounded operators, see [12, Lemma 1.4.8]. That is, as an l 2 -bounded operator-valued function (L p z) 1 is analytic on ρ(l p ) which is consistent with (L 2 z) 1 on ρ(l p ) ρ(l 2 ). Note that, (L 2 z) 1 is analytic on ρ(l 2 ) which is also the maximal domain of analyticity. Thus, the statement follows by unique continuation. Proof of Theorem 2.1. We start by showing σ(l p ) σ(l 2 ). For the kernel g (2) z of (L 2 z) 2 and fixed x,y X the function ρ(l 2 ) C, z g z (2) (x, y) is analytic. By Lemma 4.3 we know that for any compact K ρ(l 2 ) the operators (L 2 z) 2, z K are bounded on l p, p [1, ]. Therefore (L 2 z) 2 is analytic as a family of l p -bounded operators for z ρ(l 2 ). On the other hand, (L p z) 2 is analytic as a family of l p -bounded operators with domain of analyticity ρ(l p ) by [29, Lemma 3.2]. Since (L 2 z) 2 and (L p z) 2 agree on {w C Rw<0}, by unique continuation they agree as analytic l p operator-valued functions on ρ(l 2 ). As the domain of analyticity of (L p z) 2 is ρ(l p ), this implies ρ(l 2 ) ρ(l p ). On the other hand, since σ(l p ) σ(l 2 ) [0, ) the statement σ(l 2 ) σ(l p ) follows from Lemma Proof for uniformly positive measures In this section we consider measures that are uniformly bounded from below by a positive constant and prove Theorem 2.2. We notice that inf x X m(x) > 0 implies l p l q, 1 p q. Moreover, by [39, Theorem 5] we know the domains of the generators L p in this case explicitly, namely D(L p ) := { f l p Lf l p},

288 F. Bauer et al. / Advances in Mathematics 248 (2013) where L was defined in Section 2.3. In particular, this gives D(L p ) D(L q ), 1 p q. Furthermore, it can be checked directly that C c (X) D(L p ). Lemma 5.1. Assume inf x X m(x) > 0. Then, for all 1 p q and all z ρ(l p ) ρ(l q ) the resolvents (L p z) 1 and (L q z) 1 are consistent on l p = l p l q. Proof. Let 1 p<q and z ρ(l p ) ρ(l q ).AsD(L p ) D(L q ) and L p = L q on D(L p ),wehaveforallf l p l q (L q z)(l p z) 1 f = (L p z)(l p z) 1 f = f. Hence, (L p z) 1 and (L q z) 1 are consistent on l p = l p l q. Proof of Theorem 2.2. Let p [1, 2]. By the lemma above the resolvents (L p z) 1 and (L p z) 1 are consistent for z ρ(l p ) = ρ(l p ) on l p. By the Riesz Thorin interpolation theorem (L p z) 1 is bounded on l 2. We will show that (L p z) 1 is an inverse of (L 2 z) for z ρ(l p ) = ρ(l p ). So, let z ρ(l p ).AsD(L 2 ) D(L p ), l 2 l p and L 2, L p are restrictions of L we have for f D(L 2 ) (L p z) 1 (L 2 z)f = (L p z) 1 (L p z)f = f. Secondly, let f l 2 and (f n ) be such that f n l p and f n f in l 2.As(L p z) 1 is l 2 -bounded, (L p z) 1 f n (L p z) 1 f, n, in l 2. By the lemma above (L p z) 1 f n = (L p z) 1 f n D(L p ) D(L 2 ), and, therefore, (L 2 z)(l p z) 1 f n = (L p z)(l p z) 1 f n = f n f, n, in l 2. Since, L 2 is closed we infer (L p z) 1 f D(L 2 ) and (L 2 z)(l p z) 1 f = f. Hence, (L p z) 1 is an inverse of (L 2 z) and, thus, z ρ(l 2 ). Remark. The abstract reason behind Theorem 2.2 is that the semigroup e tl is ultracontractive, i.e. a bounded operator from l 2 to l, which is a consequence of e tl being a contraction on l (as Q is a Dirichlet form) and the uniform lower bound on the measure. Knowing this one can deduce by duality and interpolation that e tl is a bounded operator from l p to l q, p q (cf. [42, proof of Theorem B.1.1]) and employ the proof of [29, Proposition 2.1] or [30, Proposition 3.1]. 6. Spectral properties of normalized Laplacians In this section, we consider normalized Laplace operators that is we assume m = n, where n(x) = y X b(x,y), x X, and c Symmetries of the spectrum Recall that a graph is called bipartite if the vertex set X can be divided into two disjoint subsets X 1 and X 2 such that every edge connects a vertex in X 1 to a vertex in X 2, i.e., b(x,y) > 0fora pair (x, y) X X implies (x, y) (X 1 X 2 ) (X 2 X 1 ).

289 730 F. Bauer et al. / Advances in Mathematics 248 (2013) Theorem 6.1. Assume m = n, c 0 and let p [1, ]. Then, σ(l p ) is included in {z C z 1 1} and is symmetric with respect to R ={z C Iz = 0}, i.e., λ σ(l p ) if and only if λ σ(l p ). Moreover, if the graph is bipartite, then σ(l p ) is symmetric with respect to the line {z C: Rz = 1}, i.e., λ σ(l p ) if and only if (2 λ) σ(l p ). The proof of the second part of the theorem is based on the following lemma [12, Lemma ]. Lemma 6.2. Let A : B B be a bounded operator on a Banach space B. Then λ σ(a) if and only if at least one of the following occurs: (i) λ is an eigenvalue of A. (ii) λ is an eigenvalue of A, where A is the dual operator of A. (iii) There exists a sequence (f n ) in B with f n =1 and lim n Af n λf n =0. Proof of Theorem 6.1. Let p [1, ]. Asn/m = 1 the operator L p is bounded on l p by [38, Theorem 11] (or [27, Theorem 9.3]) with bound 2. Moreover, the operator L p can be represented as L p = I P p where P p is the transition matrix acting as P p f(x)= 1 n(x) y b(x,y)f(y). Direct calculation shows that P p p,p 1. As the spectral radius is smaller than the norm, σ(p p ) {z C z 1}.Now,I is a spectral shift by 1, so the first statement follows. The first symmetry statement follows from the fact that the integral kernel of L p is real valued. Let now X 1 and X 2 be a bipartite partition of X. For a function f : X C, we denote f = 1 X1 f 1 X2 f, where 1 W is the characteristic function of W X. We separate the proof into three cases according to the preceding lemma. Case 1: λ is an eigenvalue of L p with eigenfunction f. By direct calculation it can be checked that f is an eigenfunction of L p for the eigenvalue 2 λ. Case 2: λ is an eigenvalue of L p. By the same argument as in Case 1 we get that 2 λ is an eigenvalue of L p. Therefore, by σ(l p ) = σ(l p), we conclude the result. Case 3: λ is such that there is f n with f n =1, n 1 and lim (L p λ)f n p = 0. It is easy to see that f n p = f n p and by direct calculation it follows (L p (2 λ)) f n p = (L p λ)f n p. Thus, the statement (2 λ) σ(l p ) follows by Lemma 6.2 (iii) Cheeger constants Define the Cheeger constant α 0 to be the maximal β 0 such that for all finite W X βn(w) W, where W := (x,y) W (X\W) b(x,y). For recent developments concerning Cheeger constants see also [2,3]. Proposition 6.3. Assume m = n and c 0. Then, inf σ(l 1 ) = inf σ(l 2 ) if and only if α = 0.

290 F. Bauer et al. / Advances in Mathematics 248 (2013) Proof. As L is bounded it is easy to see that the constant functions are eigenfunctions to the eigenvalue 0. Hence, inf σ(l 1 ) = inf σ(l ) = 0. Now, 0 = inf σ(l 2 ) is equivalent to α = 0by a Cheeger inequality [38] (cf. [17] and [16]). Remark. The proposition can also be obtained as a consequence of [45, Theorem 3.1]. The considerations therein show that one direction of the result is still valid in certain situations involving unbounded operators using the Cheeger constant defined in [3]. However, in this case it is usually hard to determine whether the constant functions are in the domain of L. Nevertheless, this is the case if the graph is stochastically complete (see [39] for characterizations of this case) Spectral independence and superexponential growth This previous proposition shows that α = 0 is a necessary condition for the p-independence of the spectrum for p [1, ]. However, the next proposition shows that if we exclude p ={1, }, then we can have p-independence of the spectrum even if α>0 or if the subexponential volume growth condition is not satisfied. Define the Cheeger constant at infinity α 0 to be the maximum of all β 0 such that for some finite K X and all finite W X \ K and βn(w) W. One can easily check that under the assumption of m(x) =, α>0 if and only if α > 0: The inequality α α is obvious. If α = 0butα > 0, the Cheeger estimates, see e.g. [22,37,38] imply 0 σ(l 2 ),but0/ σ ess (L 2 ). This, however, means that 0 is an eigenvalue which is impossible as the constant functions are not in l 2 due to m(x) =,cf.[24, Theorem 6.1]. Hence, the next theorem says that we have p-independence for p (1, ) even though α>0. Theorem 6.4. Assume m = n and α = 1. Then, σ(l p ) = σ(l 2 ) for all p (1, ) and the graph has superexponential volume growth with respect to the natural graph metric, i.e., lim r 1 r log n(b r (x)) =, for all x X. Proof. It was shown in [38] (cf. [37,22] for the unweighted case) that α = 1 implies σ ess (L 2 ) ={1} which is equivalent to I L 2 being a compact operator. Since I L p is a consistent family of bounded operators for p [1, ] it follows from a well-known result by Krasnosel skii and Persson (see [12, Theorem ]) that I L p is compact for all p [2, ). Using a theorem of Schauder (see for instance [12, Theorem ]), we conclude that I L p is compact for all p (1, ). Now, it follows from [12, Theorem ] that the spectrum of L p is p-independent for all p (1, ). The second part of the proposition follows directly from [21, Theorem 1] (for the weighted case combine [38, Theorem 19] and [28, Theorem 4.1]). Remark 6.5. (a) For examples satisfying the assumptions of the theorem above see [37,22]. (b) Under the assumption m = n and α>0 one can show that I L 1 and I L are not compact operators. Assume the contrary, i.e., I L 1 and I L are compact, then [12, Theorem ] implies that the spectrum is p-independent for all p [1, ]. This, however, contradicts Proposition 6.3.

291 732 F. Bauer et al. / Advances in Mathematics 248 (2013) Tessellations and curvature For this final section, we restrict our attention to planar tessellations and relate curvature bounds to volume growth and p-independence. We show analogue statements to Propositions 1 and2of[43] and discuss the case of uniformly unbounded negative curvature. We consider graphs such that b takes values in {0, 1} and c 0. The two prominent choices for m are either m = n or m 1. For m = n the natural graph metric d n is an intrinsic metric and for m 1 the path metric d 1 given by d 1 (x, y) = inf x=x 0 x n =y n ( n(xi 1 ) n(x i ) ) 1 2, x,y X, i=1 is an intrinsic metric. Both metrics have the jump size at most 1, in particular, both metrics have finite jump size. We denote the Laplacian with respect to m = n by (n) p and with respect to m 1by p, p [1, ]. Note that by Theorem 2.2 we have for all p [1, ] σ ( (n) ) ( ) 2 σ (n) p and σ( 2 ) σ( p ). Let G be a planar tessellation, see [4,5] for background. Denote by F the set of faces and denote the degree of a face f F, that is the number of vertices contained in face, by deg(f ). We define the vertex curvature κ : X R by κ(x) = 1 n(x) 2 + f F, x f 1 deg(f ). The following theorem is a discrete analogue of [43, Proposition 1]. Theorem 7.1. If κ 0, then it has quadratic volume growth both with respect to d n and m = n and with d 1 and m 1. In particular, σ( (n) p ) = σ( (n) 2 ) and σ( p) = σ( 2 ) for all p [1, ]. Proof. By [32, Theorem 1.1] the volume growth for κ 0 is bounded quadratically with respect to the measure m = n and the metric d n. Moreover, it is direct to check that κ 0 implies that the vertex degree is bounded by 6. Hence, we have bounded geometry and thus d n and d 1 are equivalent and for m 1wehavem n/6. Thus, the volume growth of the graph is quadratically bounded also with respect to the measure m 1 and the metric d 1. The in particular is now a consequence of Theorem 2.1. Remark. The result can be easily extended to planar tessellation with finite total curvature, i.e., x X κ(x) < or equivalently vanishing curvature outside of a finite set, see [7] and also [15]. This can be seen as [32, Theorem 1.1] easily extends to the finite total curvature case. The next theorem is a discrete analogue of [43, Proposition 2]. We say that a graph with measure m has at least exponential volume growth with respect to a metric d if for the corresponding distance balls B r (x) about some vertex x, μ = lim inf r inf x X 1 r log m( B r (x) ) > 0.

292 F. Bauer et al. / Advances in Mathematics 248 (2013) Theorem 7.2. If κ<0, then the tessellation has at least exponential volume growth, both with respect to d n and m = n and with d 1 and m 1. Moreover, inf σ( (n) 1 ) inf σ( (n) 2 ) and if we have additionally bounded geometry, then inf σ( 1 ) inf σ( 2 ). Proof. [31, Theorem C, Proposition 2.1] implies that if κ(x) < 0 for all x X, then sup x X κ(x) 1/1806 < 0. Hence, by [31, Theorem B] we have positive Cheeger constant, α>0. Thus, by a Cheeger inequality [17] inf σ( (n) 2 ) α2 /2 > 0 and by elementary computations (cf. [37]) we infer inf σ( 2 ) inf σ( (n) 2 ) inf x deg(x) > 0. By [28, Corollary 4.2] this implies at least exponential volume growth with respect to both metrics. The second statement follows along the lines of the proof of Proposition 6.3. We end this section by an analogue theorem of Theorem 6.4. Theorem 7.3. If inf K X, finite sup x X\K κ(x) =, then σ( (n) p ) = σ( (n) 2 ) and σ( p) = σ( 2 ) for all p (1, ). In particular, the spectrum is purely discrete and the eigenfunctions of 2 are contained in l p for all p (1, ). Proof. By [37, Theorem 3] the curvature assumption is equivalent to pure discrete spectrum of 2. Moreover, this is equivalent to compact resolvent and compact semigroup on l 2. Thus, the statement for p follows from [8, Theorem 1.6.3]. On the other hand, the curvature assumption implies α = 1 [37], and the statement about (n) p follows from Theorem 6.4. Acknowledgments The first and second authors thank Jürgen Jost for many inspiring discussions on this topic. The first author acknowledges partial financial support of the Alexander von Humboldt Foundation and partial financial support of the NSF grant DMS Differential Equations in Geometry. The research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme (FP7/ )/ERC grant agreement No The second author thanks Zhiqin Lu for introducing him this problem and stimulating discussions at Fudan University. The third author thanks Daniel Lenz for generously sharing his knowledge on intrinsic metrics and acknowledges the financial support of the German Science Foundation (DFG), the Golda Meir Fellowship, the Israel Science Foundation (grant No. 1105/10 and No. 225/10) and BSF grant No References [1] W. Arendt, Gaussian estimates and interpolation of the spectrum in L p, Differential Integral Equations 7 (5) (1994) [2] F. Bauer, B. Hua, J. Jost, The dual Cheeger constant and spectra of infinite graphs, preprint, , [3] F. Bauer, M. Keller, R. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, J. Eur. Math. Soc. (JEMS) (2013), in press, preprint, arxiv: , [4] O. Baues, N. Peyerimhoff, Curvature and geometry of tessellating plane graphs, Discrete Comput. Geom. 25 (1) (2001) [5] O. Baues, N. Peyerimhoff, Geodesics in non-positively curved plane tessellations, Adv. Geom. 6 (2) (2006)

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295 CHAPTER 13 M. Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Mathematische Annalen 346 (2010),

296 Math. Ann. DOI /s y Mathematische Annalen The essential spectrum of the Laplacian on rapidly branching tessellations Matthias Keller Received: 29 January 2009 / Revised: 10 April 2009 Springer-Verlag 2009 Abstract In this paper, we characterize absence of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover, we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature. 1 Introduction and main results The paper is dedicated to investigate the essential spectrum of the Laplacian on graphs. More precisely the purpose is threefold. First, we give a comparison theorem for the essential spectra of the Laplacian used in the Mathematical Physics community (see for instance [1 4,8,17,18,20,23,24,29]) and the geometric Laplacian used in Spectral Geometry (see for instance [13,14,16,27]) on general graphs. Second, we consider graphs which are rapidly branching, i.e. the vertex degree is growing uniformly as the vertex tends to infinity. We establish a criterion under which absence of essential spectrum of the Laplacian is completely characterized. This criterion will be positivity of a Cheeger constant at infinity introduced in [16], based on [10,11]. It turns out that in the case of planar tessellating graphs this positivity will be implied automatically by uniform growth of vertex degree. Moreover, we can interpret the rapidly branching property as a uniform decrease of curvature. An immediate consequence is that these operators have no continuous spectrum. The third purpose is to demonstrate that and may show a very different spectral behavior. Therefore we discuss a particular class of rapidly branching graphs. This discussion will also prove independence of our assumptions in the results mentioned above. In the following introduction we will give an overview. We refer to Sect. 2 for precise definitions. M. Keller (B) Mathematical Institute, FSU Jena, Jena, Germany m.keller@uni-jena.de 123

297 M. Keller There is a result of Donnelly and Li [15] on negatively curved manifolds. It shows that the Laplacian on a rapidly curving manifold has a compact resolvent, i.e. empty essential spectrum. Theorem [Donnelly, Li] Let M be a complete, simply connected, negatively curved Riemannian manifold and K (r) = sup{k (x,π) d(p, x) r} the sectional curvature for r 0, where d is the distance function on the manifold, p M and π is a two plane in T x M. If lim r K (r) =, then on M has no essential spectrum i.e. σ ess ( ) =. A remarkable result of Fujiwara [16] provides an analogue in the graph case for the geometric Laplacian. Theorem (Fujiwara) Let G = (V, E) be an infinite graph. Then σ ess ( ) ={1} if and only if α = 1. Here α is a Cheeger constant at infinity. Since the geometric Laplacian is a bounded operator the essential spectrum can not be empty. Yet it shrinks to one point for α = 1. We will show that an analogous result holds for the Laplacian, which is used by mathematical physicists. Let G = (V, E) be an infinite graph. For finite K V denote by K c its complement V \K and let m K = inf{deg(v) v K c } and M K = sup{deg(v) v K c }, where deg : V N is called the vertex degree, which is defined as the number of edges emanating from a vertex. Denote m = lim K m K and M = lim K M K. In the next section, we will be precise about what we mean by the limits. We call a graph rapidly branching if m =. We will prove the following theorems. Theorem 1 Let G be infinite. For all λ σ ess ( ) and m inf σ ess ( ) λ M sup σ ess ( ) inf σ ess ( ) min{m, M inf σ ess ( )}. In the first statement we have the convention that if inf σ ess ( ) = 0 and m = we set m inf σ ess ( ) = 0. The first part of the theorem shows that the essential spectra of the operators and are related in terms of the minimal and maximal vertex degree at infinity. The second part gives two options to estimate the infimum of the essential spectrum of from above. Our second theorem is the characterization of absence of the essential spectrum. 123

298 The essential spectrum of the Laplacian Theorem 2 Let G = (V, E) be infinite and α > 0. Then σ ess ( ) = if and only if m =. Note that m = does not imply α > 0orσ ess ( ) =. This will be discussed in Sect. 5. We may interpret α > 0 as an assumption on the graph to be hyperbolic at infinity. (See discussion in [22] and the references [19,21,26] found there.) Moreover the growth of the vertex degree can be interpreted as the decrease of the curvature. In this way we may understand Theorem 2 as an analogue of a result of Donnelly and Li [15] for on graphs. For tessellating graphs this analogy will be even more obvious. There is a recent result of Wojciechowski [29] which proves under stronger assumptions the direction that uniform growth of the vertex degree implies absence of essential spectrum. Since the continuous spectrum of an operator is always contained in the essential spectrum there is an immediate corollary. Corollary 1 Let G = (V, E) be infinite, α > 0 and m =. Then has pure point spectrum. The class of examples for which [16] shows absence of essential spectrum are rapidly branching trees. We will show that the result is also valid for rapidly branching tessellations. We will formulate the statement in terms of the curvature because this makes the analogy to Donnelly and Li s result more obvious. For this sake we define the combinatorial curvature function κ : V R for a vertex v V as it is found in [6,7,22,28]by κ(v) = 1 deg(v) 2 + f F,v f 1 deg( f ), where the face degree deg( f ) denotes the number of vertices contained in a face f F. For finite K V let κ K = sup{κ(v) v K c } and κ = lim K κ K. Obviously κ = is equivalent to m =. Here is our main theorem. Theorem 3 Let G be a tessellation. Then σ ess ( ) = if and only if κ =. Moreover κ = implies σ ess ( ) ={1}. The theorem is a special case of Theorem 2. The hyperbolicity assumption α > 0 follows from κ = in the case of tessellating graphs. In particular it is even true that α = 1 whenever the curvature tends uniformly to. Klassert et al. [24] proved the absence of finitely supported eigenfunctions for tessellations which are non-positively curved in each corner. To assume this in our case is natural, since non-positive corner curvature is already implied by a vertex degree greater or equal 6. So under this assumption we have pure point spectrum such that all eigenfunctions are supported on an infinite set of vertices. 123

299 M. Keller The paper is structured as follows. In Sect. 2, we will define the versions of the Laplacian which appear in different contexts of the literature. We discuss Fujiwara s Theorem which can be understood as a result on compact operators. In Sect. 3 we prove Theorem 1 and 2. In Sect. 4, we give an estimate of the Cheeger constant at infinity for planar tessellations and prove Theorem 3. Finally in Sect. 5, we discuss a class of examples which shows that for general graphs and can have a quite different spectral behavior. 2 The geometric Laplacian in terms of compact operators Let G = (V, E) be a connected graph with finite vertex degree at each vertex. For a positive weight function g : V R + let { l 2 (V, g) = ϕ : V R ϕ,ϕ g = } g(v) ϕ(v) 2 <, v V c c (V ) ={ϕ : V R supp ϕ < } where supp is the support of a function. For g = 1wewritel 2 (V ). Notice that l 2 (V, g) is the completion of c c (V ) under, g.forg = deg it is clear that l 2 (V, deg) l 2 (V ) and if sup v V deg(v) < then l 2 (V ) = l 2 (V, deg). Define the operators A and D on c c (V ) by (Aϕ)(v) = u v ϕ(u) and (Dϕ)(v) = deg(v)ϕ(v). The operator A is often called the adjacency matrix. Since we assumed that the graph has no isolated vertices the operator D has a bounded inverse. The Laplace operator plays an important role in different areas of mathematics. Yet there occur different versions of it. To avoid confusion we want to discuss them briefly. We start with the Laplacian used in the Mathematical Physicist community in the context of Schrödinger operators. For reference see e.g. [8,11] (and the references therein) or in more recent publications like [1 4,12,17,18,20,23,24]. (1) The operator D A defined on c c (V ) yields the following form dϕ,dϕ = 1 ϕ(u) ϕ(v) 2. 2 v V u v The unique self adjoint extension on a subspace D( ) of l 2 (V ) of the operator corresponding to this form will be denoted by.itgivesforϕ D( ) and v V ( ϕ) (v) = deg(v)ϕ(v) u v ϕ(u). (1) Notice that is unbounded if there is no bound on the vertex degree. 123

300 The essential spectrum of the Laplacian We next introduce the geometric Laplacian. Two unitarily equivalent versions are found in the literature. They are given as follows. (2) Let = I à = I D 1 A be defined on l 2 (V, deg), where I is the identity operator. It is easy to see that is bounded and self adjoint. For ϕ l 2 (V, deg) and v V it gives ( ϕ)(v) = ϕ(v) 1 deg(v) ϕ(u). The matrix à is often called the transition matrix. This version of the geometric Laplacian can be found for instance in [13,14,16,27] and many others. Moreover is the self adjoint extension on l 2 (V, deg) of the operator associated with the form given by (1) onc c (V ). (3) There is a unitarily equivalent version as discussed e.g. in [9]. Let u v = I  = I D 1 2 AD 1 2 be defined on l 2 (V ).Itgivesforϕ l 2 (V ) and v V ( ϕ)(v) = ϕ(v) u v 1 deg(u) deg(v) ϕ(u). Notice that the operator D 1 2 1,deg : l 2 (V, deg) l 2 (V ), ϕ deg ϕ, is an isometric isomorphism and we denote its inverse by D 1 2 deg,1. Then Moreover on c c (V ) = D 1 2 1,deg D 1 2 deg,1. = D 1 2 D 1 2. Furthermore, we define the Dirichlet restrictions of these operators. For a set K V let P K : l 2 (V, g) l 2 (K c, g) be the canonical projection and i K : l 2 (K c, g) l 2 (V, g) its dual operator, which is the continuation by 0 on K. For an operator B on l 2 (V, g) we write B K = P K Bi K. 123

301 M. Keller Hence we can speak of K, K or K on K c with Dirichlet boundary conditions. In most cases K will be a finite set. For a graph G and finite K V define the Cheeger constant, see[10,14,16], α K = E W inf W K c, W < A(W ), where W denotes the cardinality of a set W, E W is the set of edges which have one vertex in W and one outside and A(W ) = v W deg(v). LetW K c,fork finite and χ the characteristic function of W. Two simple calculations, mentioned in [13] yield K χ,χ deg = K χ,χ = E W and χ,χ deg = Dχ,χ = A(W ). This gives α K = inf W K c, W < χ W χ W deg χ W,χ W deg. (2) The set K (V ) of finite subsets of V is a net under the partial order. We say that a function F : K (V ) R, K F K converges to F R if for all ɛ > 0 there is a K ɛ K (V ) such that F K F <ɛfor all K K ɛ. We then write lim K F K = F. With this convention we define the Cheeger constant at infinity as in [16] by α = lim K α K. The limit always exists since α K α L 1 for finite K L V. Therefore we can think of taking the limit over distance balls of an arbitrary vertex. The next part is dedicated to a discussion of [16]. We will look at the result from the perspective of compact operators. The proof is based on two propositions which hold for general graphs. We present them here as norm estimates on the transition matrix. As shown in [13] infσ( ) α. This easily leads to the following statement. Proposition 1 For any finite set K V Ã K 1 α K. In particular, inf σ( K ) α K. The second proposition is derived from Proposition 1 in [16] which is inspired by a calculation in [14]. 123

302 The essential spectrum of the Laplacian Proposition 2 For any finite set K V ÃK 1 α 2 K. The next theorem is direct consequence of the two propositions above and can be found in [16]. Theorem 4 For K V finite 1 1 α 2 K K α 2 K. The essential parts (ii) (v), (ii) (iii) of the next theorem are already found in [16]. The remaining statements are minor extensions. Knowing Proposition 1 and 2 one only needs standard methods for compact operators (and Propositions 3 in Sect. 3) to prove the following theorem. Theorem 5 Let G be infinite. The following are equivalent. (i) σ ess ( ) consists of one point. (ii) σ ess ( ) ={1}. (iii) Ã is compact. (iv) lim K Ã K =0. (v) α = 1. 3 The essential spectrum of In this section, we compare the operators and. We will establish bounds on the essential spectrum of by using bounds obtained for. Therefore, we will firstly state a well known proposition which allows us to determine the essential spectrum of an operator via its restriction on the complement of larger and larger sets. Then we prove two propositions which estimate the infimum of the essential spectrum of from below and above. These will be the ingredients for the proofs of Theorem 1 and 2. Proposition 3 Let G = (V, E) be infinite and B a self adjoint operator which is bounded from below and c c (V ) dense in D(B) l 2 (V, g) w.r.t. to the graph norm. Then inf σ ess (B) = sup σ ess (B) lim K lim K Bϕ,ϕ g inf = lim ϕ c c (V ) ϕ,ϕ inf σ(b K ), g K supp ϕ K c sup Bϕ,ϕ g ϕ c c (V ) ϕ,ϕ g = lim K ). K supp ϕ K c If B is bounded, we have equality in the second formula. 123

303 M. Keller The proof follows by standard techniques, that is why we omit it here. (Note that one uses lim K E],λ] i K = 0forλ<inf σess (B), where E ],λ] is the spectral projection of B.) Since and are unitarily equivalent it makes no difference whether we compare the operators and or the operators and.yet and are defined on the same Hilbert space, so it seems to be notationally easier to compare them. However to do this the following identity is vital. For ϕ c c (K c ) one can calculate K ϕ,ϕ ϕ,ϕ = D 1 2 K K D 1 2 K ϕ,ϕ ϕ,ϕ = K D 1 2 K ϕ, D 1 2 K ϕ D 1 2 K ϕ, D 1 2 K ϕ D K ϕ,ϕ. (3) ϕ,ϕ Proposition 4 Let G be an infinite graph. Then for λ σ ess ( ) m inf σ ess ( ) λ M sup σ ess ( ). Proof Let K V be finite. By Eq. 3 we have for ϕ c c (K c ) K ϕ,ϕ ϕ,ϕ K D 1 2 K ϕ, D 1 2 K ϕ D 1 2 K ϕ, D 1 2 K ϕ inf v supp ϕ deg(v). For every ψ c c (K c ) there is an ϕ c c (K c ) such that ψ = D 1 2 K ϕ. Furthermore c c (K c ) is dense in the domain of K which allows us to conclude inf σ ess ( ) = inf σ ess ( K ) m inf σ( K ). By Proposition 3 this yields the lower bound. If M = the upper bound is infinity. Otherwise by Eq. 3 sup σ( K ) M sup σ( K ) and again by Proposition 3 we have the upper bound. Proposition 5 Let G be an infinite graph. Then inf σ ess ( ) min{m, M inf σ ess ( )}. Proof Let v n V, n N be pairwise distinct such that deg(v n ) m. Moreover let χ n the characteristic function of v n.fork finite such that v n K c we have K ϕ,ϕ inf ϕ c c (K c ) ϕ,ϕ K χ n,χ n = deg(v n ) m. By Proposition 3 we have inf σ ess ( ) m. On the other hand we have by Eq. 3 inf σ( K ) M K inf σ( K ). By Proposition 3 we get inf σ ess ( ) M inf σ ess ( ). 123

304 The essential spectrum of the Laplacian Proof of Theorem 1 Recall that the operators and are unitarily equivalent. Thus σ ess ( ) = σ ess ( ). The Theorem follows from Propositions 4 and 5. Proof of Theorem 2 By Theorem 4 we have inf σ( K ) 1 1 α 2 K 0. Thus by taking the limits Proposition 3 yields inf σ ess ( ) > 0ifα > 0. Propositions 4 and 5 give the desired result. 4 Rapidly branching tessellations Examples discussed in [16] are rapidly branching trees. Fujiwara showed that for trees α = 1isimpliedbym =. Therefore by Theorem 2 we have σ ess ( ) = in the case of trees. In this section, we want to extend the class of examples to tessellations. We do this by showing that α = 1isimpliedbym = for tessellations as well. For planar graphs tessellations are quite well understood. We only recall the definitions and refer the reader to [6,7] and the references contained therein. Let G = (V, E) be a planar, locally finite graph without loops and multiple edges, embedded in R 2. We denote the set of closures of the connected components in R 2 \ e E e by F and refer to the elements of F as faces of G. We may write G = (V, E, F). A union of faces is called a polygon if it is homeomorphic to a closed disc in R 2 and its boundary is a closed path of edges without repeated vertices. The graph G is called a tessellation or tessellating if the following conditions are fulfilled. i) Any edge is contained in precisely two different faces. ii) Two faces are either disjoint or intersect in a unique edge or vertex. iii) All faces are polygons. Note that a planar tessellating graph is always infinite. From now on let G = (V, E, F) be tessellating and we do not distinguish between the graph and its embedding in R 2. Recall that the vertex degree deg(v) is the number of edges emanating from a vertex v and the face degree deg( f ) is the number of vertices contained in a face f. For a set W V let G W = (W, E W, F W ) be the induced subgraph, which is the graph with vertex set W and the edges of E which have two vertices in W. Euler s formula states for a connected finite subgraph G W W E W + F W =2. (4) Observe that the 2 on the right hand side occurs since we also count the unbounded face. Euler s formula is a part of mathematical folklore. A proof can be found for instance in [5]. We denote by F W the set of faces in F which contain an edge of E W.In fact each face in F W contains at least two edges in E W. Therefore F W E W follows easily. Moreover we define for finite W V the inner degree of a face f F by deg W ( f ) = {v W v f } 123

305 M. Keller Finally let C(W ) be the number of connected components in G V \W. Loosely speaking it is the number of holes in G W. We need two important formulas which hold for arbitrary finite subgraphs G W = (W, E W, F W ) of G. Recall A(W ) = v W deg(v). The first formula can be easily rechecked. It reads A(W ) = 2 E W + E W. (5) As for the second formula note that F W has faces which are not in F. Nevertheless F W C(W ) = F W F. This is the number of bounded faces which are enclosed by edges of E W. Thus sorting the following sum over vertices according to faces gives the second formula v W f F, f v 1 deg( f ) = F W C(W ) + f F W deg W ( f ) deg( f ). (6) Lemma 1 Let G = (V, E, F) be a tessellating graph. Then for a finite and connected set W V E W A(W ) 6( W +C(W ) 2). Proof By the tessellating property we have f F W deg W ( f ) E W. Moreover deg( f ) 3for f F. Combining this with Eq. 6 we obtain F W 1 3 v W 1 f v f F W 1 3 (A(W ) E W ) + C(W ). deg W ( f ) + C(W ) Applying Euler s formula (4) and equation (5) to the left hand side of the inequality above, we obtain 2 W 1 6 (A(W ) E W ) + C(W ), which yields the Lemma. Now we give an estimate from below of the Cheeger constant at infinity. 123

306 The essential spectrum of the Laplacian Proposition 6 Let G = (V, E, F) be a tessellating graph. Then α 1 lim K sup v K c 6 deg(v). Proof We assume w.l.o.g. that the finite sets K are distance balls. To calculate α K we can restrict ourselves to finite sets W V, which are connected. Otherwise we find a connected component W 0 of W such that E W 0 /A(W 0 ) E W /A(W ). Moreover we can choose W such that C(W ) 2. Otherwise we find a superset W 1 of W such that E W 1 /A(W 1 ) E W /A(W ). Obviously A(W ) W inf v W deg(v). By Lemma 1 E W A(W ) A(W ) 6 W A(W ) 1 6 inf v W deg(v). Hence we have α K 1 sup v K c 6/ deg(v). We obtain the result by taking the limit over all finite sets. Remark 1 The relation between curvature and the Cheeger constant can be presented in more detail than we need it for our purpose here. See therefore [25]. Proof of Theorem 3 Let κ =. This is obviously equivalent to m = which implies α = 1 by Proposition 6. Thus by Theorems 5 and 1 we obtain σ ess ( ) ={1} and σ ess ( ) =. On the other hand Proposition 5 tells us that σ ess ( ) = implies m = and thus κ =. Remark 2 The implication that σ ess ( ) = follows from κ = can also be obtained in an alternative way. Higuchi [22] and Woess [28] showed independently that α K > 0 whenever κ K < 0forK =. Since m = is implied by κ = we can apply Theorem 1 immediately. 5 A further class of rapidly branching graphs In this section, we want to discuss a class of examples which demonstrates that and can show very different spectral phenomena. In particular these examples prove the independence of our assumptions in Theorem 2. Let G(n) = (V (n), E(n)) be the full graph with n vertices. For γ 0 and c 1 let N γ,c : N N, n n[cn γ ], where [x] is the the smallest integer bigger than x R. Denote N 1 = 1, N 2 = max{[c], 2} and for k 3 N k = N γ,c (N k 1 ). 123

307 M. Keller We construct the graph G γ,c as follows. We start by connecting the vertex in G(N 1 ) with each vertex in G(N 2 ). We proceed by connecting each vertex in G(N k ) uniquely with [cn γ k ] vertices in G(N k+1) for k N. Obviously G γ,c is rapidly branching whenever γ > 0orc > 1, in fact N k 2 k 1. From another point of view G γ,c is a tree of branching number [cn γ k ] in the kth generation, where we connected the vertices of each generation with one another. The next theorem shows a scheme of the quite different behavior of the sets σ ess ( ) and σ ess ( ) for the graphs G γ,c. Theorem 6 For γ 0 and c 1 let G γ,c be as above. (1) If γ = 0, then α = 0, inf σ ess ( ) = 0 and inf σ ess ( ) [c]. (2) If γ ]0, 1[, then α = 0, inf σ ess ( ) = 0 and σ ess ( ) =. (3) If γ = 1, then α = 1+c c ess( ) ]0, 1[ and σ ess ( ) =. (4) If γ>1, then α = 1, σ ess ( ) ={1} and σ ess ( ) =. As mentioned above all graphs G γ,c are rapidly branching if γ > 0orc > 1. The theorem shows the independence of our assumptions and thus optimality of the result. More precisely the case γ = 0 shows that m = alone does not imply σ ess ( ) =. On the other hand the case γ ]0, 1[ makes clear that σ ess ( ) = does not imply α > 0. Moreover when γ = 1 we see that m = and α > 0 does not imply α = 1. The last case is an example where σ ess ( ) ={1} and σ ess ( ) = as in the case of trees and tessellations. For a graph G denote by B n the set of vertices which have distance n N or less from a fixed vertex v 0 V. In our context choose v 0 as the unique vertex in G(N 1 ). The intuition behind the theorem is as follows. Let S n,k = B n \B k, n > k and χ = χ Sn,k its characteristic function. Then one can calculate Bk χ,χ χ,χ = E S n,k A(S n,k ) A(S n,k ) S n,k c N 1 γ n + c N γ n ( N 1 γ n ) + c = cnn γ. The left hand side may thought to be close to inf σ( Bk ). Moreover the first factor after the equality sign may thought to be close to α Bk. If this is true we can control the growth and the decrease of these terms by γ. For instance inf σ( Bk ) would increase to infinity although α Bk tends to zero for γ<1. We denote for a vertex v B k deg ± (v) = {w S k±1 v w}, where we set S k = B k \B k 1 for k 2. To prove the theorem we will need the following three Lemmata. Lemma 1.5 of [13] and the remark that follows it yield the following. Lemma 2 Let G be a graph. If (deg + (v) deg (v))/ deg(v) C for all v B c n then α Bn C. With the help of this Lemma we will prove the statements for α on the respective graphs. 123

308 The essential spectrum of the Laplacian Lemma 3 Let c If γ<1then α = If γ = 1 then α = 1+c c. 3. If γ>1then α = 1. Proof We get an estimate from above by calculating α Bn 1 E S n A(S n ) = [ γ ] N n cnn + Nn [ γ ] = N [ γ ] n cnn + Nn N n (N n 1) + N n cnn + Nn Nn 2 + N [ γ ]. n cnn To obtain a lower bound for α Bn 1 we use Lemma 2 and calculate deg inf + (v) deg (v) v Bn 1 c deg(v) = inf k n [ γ ] cn k N k + [ cn γ ] = k [ γ ] cnn N n + [ cnn γ ]. One gets the desired result by letting n tend to infinity. The next lemma is crucial to show absence of essential spectrum for when γ>0. Lemma 4 Let γ > 0 and ϕ k functions in c c (Bk 1 c ) such that ϕ k 1 and Bk 1 ϕ k,ϕ k C for all k N and some constant C > 0. Then lim k ϕ k = 0. Proof Choose ϕ k, k N as assumed. Denote by ϕ (i) k the restriction of ϕ k to S i = B i \B i 1 for i k and choose m > k such that supp ϕ k B m. Then an estimate on the form of Bk 1 reads m Bk 1 ϕ k,ϕ k i=k v S i w S i+1,w v m v Si i=k 2 v S i ϕ k (v) ϕ k (w) 2 [ γ ] cn i ϕ 2 k (v) + ϕk 2 (w) w S i+1 w S i+1,w v ϕ k (v)ϕ k (w) 123

309 M. Keller = m [ cn γ ] i=k 2 m i=k i [ cn γ i ( [cn γ i v S i ϕ 2 k (v) + w S i+1 ϕ 2 k (w) 1 2 ] ϕk 2 (v) ϕk 2 (w) v S i w S i+1 ] 1 2 ϕ (i) k ϕ (i+1) k ) In the second step we used that each vertex in S i is uniquely adjacent to [cn γ i ] vertices in S i+1 for k i m and in the third step we used the Cauchy Schwarz inequality. We assumed Bk 1 ϕ k,ϕ k C and in particular this is true for every term in sum we estimated above. Moreover ϕ k 1fork i m and thus ϕ (i) k ϕ (i+1) k ϕ C + (i+1) k C + 1 c 2 1 γ 2 N c 2 1 γ. 2 N Set C 0 = ( C + 1)/c 1 2. Since the sequence i i ( ) N γ 2 i is summable we deduce ϕ k m ϕ (i) k i=k C 0 m i=k N γ 2 i C 0 i=k N γ 2 i <. We now let k tend to infinity and conclude lim k ϕ k = 0. Proof of Theorem 6 From Propositions 1, 2 and 3 we can deduce 1 1 α 2 inf σ ess( ) α. Thus by Lemma 3 we get the assertion for α and inf σ ess ( ). If γ = 0 we get for the characteristic function χ = χ Sn,k of S n,k = B n \B k, n > k Bk χ,χ χ,χ = [c](n k + N n ) ni=k+1 N i = ( ) [c] Nk N n n 1 i=k+1 N. i N n Hence by taking the limit over n we have by Proposition 3 that inf σ ess ( ) [c]. Let γ>0and let ϕ k be functions in c c (Bk+1 c ) such that ϕ k = 1 and lim Bk+1 ϕ k,ϕ k = inf σess ( ). k 123

310 The essential spectrum of the Laplacian This is possible by Proposition 3 and a diagonal sequence argument. As ϕ k = 1by Lemma 4 the term Bk ϕ k,ϕ k tends to infinity. Thus the essential spectrum of is empty. Acknowledgments I take this chance to express my gratitude to Daniel Lenz for all the fruitful discussions, helpful suggestions and his guidance during this work. I also like to thank Norbert Peyerimhoff for all helpful hints during my visit in Durham. This work was partially supported by the German Research Council (DFG) and partially by the German Business Foundation (sdw). References 1. Allard, C., Froese, R.: A Mourre estimate for a Schrödinger operator on a binary tree. Rev. Math. Phys. 12(12), (2000) 2. Aizenman, M., Sims, R., Warzel, S.: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Prob. Theory Relat. Fields 136, (2006) 3. Antunovic, T., Veselic, I.: Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs. In: Proceedings of Operator Theory, Analysis and Mathematical Physics, Lund, pp (2006) 4. Breuer, J.: Singular continuous spectrum for the Laplacian on certain sparse trees. Commun. Math. Phys. 269(3), (2007) 5. Bollobás, B.: Graph Theory: An Introductory Course. Springer-Verlag (1979) 6. Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25 (2001) 7. Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), (2006) 8. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators, Springer Verlag (1987) 9. Chung, F.: Spectral graph theory. CBMS Regional Conference Series in Mathematics, No. 92, American Mathematical Society, New York (1997) 10. Cheeger, J.: A lower bound for the lowest eigenvalue of the Laplacian. In: Problems in Analysis. A Symposium in honor of S. Bochner, pp Princeton University Press, Princeton (1970) 11. Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks. Trans. Am. Math. Soc. 284, (1984) 12. Dodziuk, J.: Elliptic operators on infinite graphs. In: Proceedings of the Conference Krzysztof Wojciechowski 50 years Analysis and Geometry of Boundary Value Problems, Roskilde, Denmark (to appear) Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Durrett, R., Pinsky, M.A. (eds.) Geometry of Random Motion, vol. 73, pp AMS Contemporary Mathematics, New York (1988) 14. Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. In: Elworthy, K.D. (ed.) From Local Times to Global Geometry, Control and Physics. Longman Scientific and Technical, pp (1986) 15. Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math J. 46, (1979) 16. Fujiwara, K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), (1996) 17. Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230, (2006) 18. Georgescu, V., Golénia, S.: Isometries, fock spaces and spectral analysis of Schrödinger operators on trees. J. Funct. Anal. 227, (2005) 19. Ghys, E., de la Harpe, P. (ed.): Sur les groupes hyperboliques d après Mikhael Gromov. Progress in Mathematics 83, Birkhäuser, Basel (1990) 20. Golénia, S.: C*-algebra of anisotropic Schrödinger operators on trees. Annales Henri Poincaré 5(6), (2004) 21. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory, pp M.S.R.I. Publ. 8, Springer, Heidelberg (1987) 22. Higuchi, Y.: Combinatorial curvature for planar graphs. J. Graph Theory (2001) 123

311 M. Keller 23. Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, (1994) 24. Klassert, S., Lenz, D., Peyerimhoff, N., Stollmann, P.: Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proc. AMS 134, No. 5 (2005) 25. Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. arxiv: Lyndon, R.S., Schupp, P.E.: Combinatorial group theory, Springer (1977) 27. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge University Press (2000) 28. Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124, (1998) 29. Wojciechowski, R.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. (to appear) 123

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313 CHAPTER 14 M. Keller, N. Peyerimhoff, Cheeger constants, growth and spectrum of locally tessellating planar graphs, Mathematische Zeitschrift 268 (2011),

314 Math. Z. (2011) 268: DOI /s Mathematische Zeitschrift Cheeger constants, growth and spectrum of locally tessellating planar graphs Matthias Keller NorbertPeyerimhoff Received: 28 August 2009 / Accepted: 8 February 2010 / Published online: 20 March 2010 Springer-Verlag 2010 Abstract In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications. 1 Introduction A locally tessellating planar graph G is a tiling of the plane with all faces to be polygons with finitely or infinitely many boundary edges. The edges of G are continuous rectifiable curves without self-intersections. Faces with infinitely many boundary edges are called infinigons and occur, e.g., in the case of planar trees. The sets of vertices, edges and faces of G are denoted by V, E and F (see the beginning of Sect. 2 for precise definitions). The function d(v, w) denotes the combinatorial distance between two vertices v, w V, where each edge is assumed to have combinatorial length one. For any pair v, w of adjacent vertices we write v w. Useful local concepts of a planar graph G are combinatorial curvature notions. The finest curvature is defined on the corners of G. A corner is a pair (v, f ) V F, wherev is a vertex of the face f.thecorner curvature κ C is defined as κ C (v, f ) = 1 v + 1 f 1 2, M. Keller (B) Mathematical Institute, Friedrich-Schiller-University Jena, Jena, Germany m.keller@uni-jena.de N. Peyerimhoff Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK norbert.peyerimhoff@durham.ac.uk 123

315 872 M. Keller, N. Peyerimhoff where v and f denote the degree of the vertex v and the face f.if f is an infinigon, we set f = and 1/ f =0. The curvature at a vertex v V is given by κ(v) = f :v f κ C (v, f ) = 1 v 2 + f :v f 1 f. For a finite set W V we define κ(w ) = v W κ(v). These combinatorial curvature definitions arise naturally from considerations of the Euler characteristic and tessellations of closed surfaces, and they allow to prove a combinatorial Gauß-Bonnet formula (see [2, Theorem 1.4]). Similar combinatorial curvature notions have been introduced by many other authors, e.g. [14,16,28,31]. Let us mention two global geometric consequences of the curvature sign: In [5] it was proved that strictly positive vertex curvature implies finiteness of a graph, thus proving a conjecture of Higuchi (which is a discrete analogue of Bonnet Myers theorem in Riemannian geometry). This question was investigated before by Stone [28]. The cut locus Cut(v) of a vertex v consists of all vertices w V,atwhichd v := d(v, ) attains a local maximum, i.e., we have w Cut(v) if d v (w ) d v (w) for all w w. If G is a plane tessellation with non-positive corner curvature, then G is without cut locus, i.e., we have Cut(v) = for all v V. This fact can be considered as a combinatorial analogue of the Cartan Hadamard theorem (for a proof and more details see [3, Theorem 1]). For a finite subset W V,letvol(W) = v W v. We consider the following two types of Cheeger constants: α(g) := inf W V, W < E W W and α(g) := inf W V, W < E W vol(w ), (1) where E W is the set of all edges e E connecting a vertex in W with a vertex in V \W. The quantity α(g) is called the physicalcheegerconstantand α(g) the geometric Cheeger constant of the graph G. The attributes physical and geometric are motivated by the fact that these constants are closely linked to two types of Laplacians (see, e.g. [21,30]) and that the first type is used in the community of Mathematical Physics whereas the second appears frequently in the context of Spectral Geometry. Cheeger constants are invariants of the global asymptotic geometry. They are important geometric tools for spectral considerations (both in setting of graphs and of Riemannian manifolds) and play a prominent role in the topic of expanders and Ramanujan graphs (see [20] for a very recommendable survey on this topic). Natural model spaces are the (p, q)-regular plane tessellations G p,q : every vertex in G p,q has degree p and every face has degree q. (In the case 1 p + q 1 < 2 1, G p,q can be realized as a regular tessellation of the Poincaré disc model of the hyperbolic plane by translates of a regular compact polygon.) The graphs G p,q can be considered as discrete counterparts of constant curvature space forms in Riemannian Geometry. The Cheeger constants of these regular graphs are explicitly given: Theorem (see [15,18,19]) Let 1 p + 1 q 1 2.Then 123 α(g p,q ) = p 2 1 p 4 (p 2)(q 2).

316 Cheeger constants, growth and spectrum of locally tessellating planar graphs 873 Let us now leave the situation of regular tessellations. It is known that the Cheeger constants of general negatively curved planar graphs are strictly positive (see [6,16,31]). Moreover, for infinite planar graphs G with v p and f q for almost all vertices and faces and c := p q 1 > 0, the following estimate was shown in [26]: α(g) 2pqc 3q 8. (2) Next we introduce a bit of notation before we state our explicit lower Cheeger constant estimates. The variables p, q in this paper always represent a pair of numbers 3 p, q satisfying 1 p + q (note that we use 1/ =0). For such a pair (p, q), let 1, if q =, Then we have C p,q := 1 + ( 2 q 2, q 2 )( 1 + if q < and p =, ) 2 (p 2)(q 2) 2, if p, q <. Theorem 1 (Cheeger constant estimate) Let G = (V, E, F) be a locally tessellating planar graph such that v p for all v V and f q for all f F. (Note that p = or q = means no condition on the vertex of face degrees.) Let C p,q be defined as in (3). (a) Assume that C := inf v V κ(v) is strictly positive. Then α(g) 2C p,q C. (b) Assume that c := inf v V v 1 κ(v) is strictly positive. Then α(g) 2C p,q c. The above estimates are sharp in the case of regular trees (in which case q = ). The proof of this theorem is given in Sect. 2. Observe that the constant C p,q 1in(3) becomes largest if the graph G in Theorem 1 has both finite upper bounds for vertex and face q( p 2) degrees. (A shorter expression for C p,q is (p 2)(q 2) 2, which we have to interpret in the right way if q = or p =.) Let us study our estimate in the regular case G = G p,q : In this case our estimate yields ( ) 2 (p 2) 1 α(g p,q ). (p 2)(q 2) 2 On the other hand, a straightforward calculation leads to the following upper bound ( ) 4 2 α(g p,q ) = (p 2) 1 (p 2) 1, (p 2)(q 2) (p 2)(q 2) 1 which shows that our lower bound is very close to the correct value. Mohar s estimate (2) in this situation coincides with ours in the particular case (p, q) = (, 3), and becomes considerably weaker for q 4orp <. Remark Every infinite connected graph G = (V, E) with v p has physical Cheeger constant α(g) p 2. To see this, choose an infinite path v 0,v 1,v 2,... and let W n := {v 0,v 1,...,v n }.Thenwehave E W n W n 2(p 1) + (n 1)(p 2), n + 1 (3) 123

317 874 M. Keller, N. Peyerimhoff which implies E W n α(g) lim n W n = p 2. Thesameargumentsshowα(T p ) = p 2and α(t p ) = p 2 p,wheret p denotes the p-regular infinite tree. Next, we turn to other global asymptotic invariants related to the growth of an infinite graph G = (V, E).Forafixedcenterv 0 V,letB n = B n (v 0 ) = {v V d(v 0,v) n} be the balls, S n = S n (v 0 ) = {v V d(v 0,v)= n} be the spheres of radius n and σ n = S n. There are two versions of the exponential growth defined as μ(g,v 0 ) := lim sup n 1 n log S 1 n and μ(g,v 0 ) := lim sup n n log vol(b n). Whenever there is a uniform bound on the vertex degree i.e., p = sup v V v < one easily checks μ(g,v 0 ) = μ(g,v 0 ). In many cases the exponential growth does not depend on the choice of v 0. If this is the case, we simply write μ(g) and μ(g). The growth series for (G,v 0 ) is the formal power series f G,v0 (z) = n=0 σ n z n.by Cauchy Hadamard criterion, the growth series represents a well-defined function in the open complex ball of radius e μ(g,v0). Of particular importance in the study of the growth series f G,v0 are recursion formulas for the sequence σ n. In this paper, we consider the case of q-face regular plane tessellations G = (V, E, F) i.e., all faces are q-gons that is f =q for all f F. Our result will be stated in terms of of normalized average curvatures over spheres ( ) 2q 1 κ n := κ(s n ) := q 2 S n κ(s n). Note that the constant 2π(q 2)/2q is the internal angle of a regular q-gon. Before stating our result we need some more notation: For 3 q < let N = q 2 2 if q is even and N = q 2ifq is odd, and 4 q 2 if q is even, 4 b l = q 2 if q is odd and l = N 1 2, (4) 4 q 2 2 N 1 ifq is odd and l = 2, for 0 l N 1. Theorem 2 (Growth recursion formulas) Let G = (V, E, F) be a q-face regular plane tessellation without cut locus, S n = S n (v 0 ) for some v 0 V and σ n = S n.letκ n, N and b l be defined as above. Then we have the following (N + 1)-step recursion formulas for n 1: σ 1 + n 1 l=0 (b l κ n l )σ n l if n < N, σ n+1 = N 1 l=0 (b l κ N l )σ N l if n = N, σ n N + (5) N 1 l=0 (b l κ n l )σ n l if n > N. 123

318 Cheeger constants, growth and spectrum of locally tessellating planar graphs 875 A proof of this theorem is given in Sect. 3. Note that the constants κ k are zero for the regular flat tessellations G 3,6, G 4,4 and G 6,3. The constants κ k in (5) can, therefore, be considered as curvature correction terms for general non-flat tessellations. In the special case of (p, q)-regular graphs G = G p,q, the terms b l κ k all coincide with the constant p 2 except in the case if q is odd and l = N 1 2,whenwehaveb l κ k = p 4. In this case, Theorem 2 is equivalent to the fact that f G,v0 g p,q = h p,q with if q is even, and h p,q (z) = 1 + 2z + +2z N + z N+1, g p,q (z) = 1 (p 2)z (p 2)z N + z N+1, h p,q (z) = 1 + 2z + +2z N z N z N z N + z N+1, g p,q (z) = 1 (p 2)z (p 4)z N+1 2 (p 2)z N + z N+1, if q is odd. This agrees with results of Cannon and Wagreich [4] and Floyd and Plotnick [10, Sect. 3] that the growth function f G,v0 is the rational function h p,q /g p,q. Moreover, it was shown in [1,4] that the denominator polynomial g p,q for 1 p + q 1 < 2 1 is a reciprocal Salem polynomial, i.e., its roots lie on the complex unit circle except for two positive reciprocal real 1 zeros x p,q < 1 < x p,q < p 1. This implies that the exponential growth coincides with log x p,q, i.e., μ(g p,q ) = μ(g p,q ) = log x p,q < log(p 1) = μ(t p ). (6) (An even more precise description of the growth of the sequence σ n isgivenin[1, Cor.3].) Of course, it is desirable to know more about the explicit value of μ(g p,q ) = log x p,q.since g p,q is divisible by z 2 (p q 2 4 )z + 1 in the case q = 3, 4, 6, we have Proposition 1.1 Let q {3, 4, 6}. Then μ(g p,q ) = μ(g p,q ) = log p 2 2 q 2 + ( p 2 2 q 2 ) 2 1. In most of the cases the polynomial g p,q is essentially irreducible (expect for some small well known factors; see [1, Thm. 1]) and there is no hope to have an explicit expression for its largest zero x p,q > 1. A direct consequence of the isoperimetric inequality in [2,Cor.5.2] is the following lower estimate of log x p,q : Proposition 1.2 Let G p,q be non-positively curved, i.e., for all v V ( 1 C = κ(v) = p p + 1 q 1 ) 0, 2 then we have ( μ(g p,q ) = μ(g p,q ) = log x p,q log 1 + 2q ) q 1 C. (7) Note that (7) implies lim q μ(g p,q ) = μ(t p ) = log p 1forallp

319 876 M. Keller, N. Peyerimhoff Remark The Mahler measure M(g) of a monic polynomial g Z[z] with integer coefficients is given by the product z i,wherez i C are the roots of g of modulus 1. Lehmer s conjecture states that for every such g with M(g) >1wehave ( M(g) M 1 z + z 3 z 4 + z 5 z 6 + z 7 z 9 + z 10) Thus (7) yields an explicit lower estimate for the Mahler measure of the polynomials g p,q. Let us now return to general q-face regular tessellations. We conclude from Theorem 2: Theorem 3 (Curvature/Growth comparison) Let G = (V, E, F) and G = (Ṽ, Ẽ, F) be two q-face regular plane tessellations with non-positive vertex curvature, S n V and S n Ṽ be spheres with respect to the centers v 0 V and ṽ 0 Ṽ, respectively, and σ n = S n and σ n = S n. Assume that the normalized average spherical curvatures satisfy κ( S n ) κ(s n ) 0, for all n 0. Then the difference sequence σ n σ n 0 is monotone non-decreasing and, in particular, we have μ( G) μ(g). In the case where there is a uniform bound on the vertex degree we also have μ( G) μ(g). A proof of this theorem is given in Sect. 3. (In fact, the proof shows that the vertex curvature conditions in Theorem 3 can be slightly relaxed: Namely, it suffices that G has non-positive vertex curvature and that both graphs G and G are without cut-loci.) This theorem can be considered as a refined discrete counterpart of the Bishop Günther Gromov comparison theorem for Riemannian manifolds (see, e.g. [13, Theorem 3.101]). The latter compares volumes of balls in Riemannian manifolds against constant curvature space forms; our discrete counterpart deals with spheres (the result for balls is obtained by adding over spheres) and is more flexible as it allows to use more general comparison spaces. If we drop the face regularity condition, it is not difficult to derive the following growth comparison with G replaced by the p-regular tree. The statement holds for arbitrary graphs. A proof of this result is given in Sect. 3: Theorem 4 (Tree comparison) Let G be a locally finite, connected graph satisfying v p for all v V,forsome p 3.Then where T p is a p-regular tree. μ(g) = μ(g) μ(t p ) = log(p 1), Note that another more involved comparison with a tree was obtained by Higuchi [17] for infinite vertex-regular graphs with each vertex contained a cycle of uniformly bounded length. Let us finally discuss some spectral applications (see [27,32] for classical surveys). The (geometric) Laplacian : l 2 (V, m) l 2 (V, m) with m(v) = v for v V is given by ( ϕ ) (v) = 1 (ϕ(v) ϕ(w)), v w v for all ϕ l 2 (V, m), v V. The relation between the bottom λ 0 (G) of the spectrum and the bottom λ ess 0 (G) of the essential spectrum of, the Cheeger constant, and the exponential growth in the discrete 123

320 Cheeger constants, growth and spectrum of locally tessellating planar graphs 877 case was presented first by Dodziuk and Kendall [8] and Dodziuk and Karp [7]. The best estimates are due to Fujiwara (see [11,12]): 1 1 α(g) 2 λ 0 (G) λ ess 0 2e μ(g)/2 (G) 1, (8) 1 + e μ(g) Note that the estimates are sharp in the case of regular trees. An immediate consequence of Theorem 1 and (8) is the following combinatorial analogue of McKean s theorem (see [25] for the result in the smooth setting): Corollary 1.3 Let G = (V, E, F) be a locally tessellating planar graph satisfying the vertex and face degree bounds in Theorem 1. Moreover, assume that c := inf v V v 1 κ(v) > 0. Then 1 1 ( 2C p,q c ) 2 λ0 (G), where C p,q is defined in (3). This estimate is sharp in the case of regular trees. Similarly, Theorem 3,(6)and(8) directly imply Corollary 1.4 Let p, q 3 and 1 p + q Let G be a q-face regular tessellation without cut locus and satisfying v p for all vertices. Then λ ess 0 (G) 1 2 x p,q. 1 + x p,q This estimate is sharp in the case of regular trees. Let us finish this introduction with some general references. It was shown in [24] that non-positive corner curvature implies non-existence of finitely supported eigenfunctions of all elliptic operators on planar graphs. Lower bounds for the bottom of the essential spectrum in terms of Cheeger constants at infinity or branching rates of general non-planar graphs can be found in [12,29] for the geometric Laplacian and in [21,33,34] for the physical Laplacian : D( ) l 2 (V ) l 2 (V ) given by ( ϕ)(v) = w v (ϕ(v) ϕ(w)), ϕ D( ), v V. These results show, in particular, the absence of the essential spectrum for graphs with curvature converging to negative infinity on complements of finite sets of an increasing exhaustion of G, a phenomenon which was first proved in the context of smooth Riemannian manifolds by [9]. The different types of Laplacians emerging in this context are due to the choice of the measure in the corresponding l 2 space. For a discussion of the general framework of these operators and recent generalizations of the results mentioned above we refer to [22,23]. 2 Proof of Theorem 1 Let us first give precise definitions of some notions used in the introduction. Let G = (V, E) be a planar graph embedded in R 2. The faces f of G are the closures of the connected components in R 2 \ e E e. We further assume that G has no loops, no multiple edges and no vertices of degree one (leaves). Moreover, we assume that every vertex has finite degree and that every bounded open set in R 2 meets only finitely many faces of G. We call a planar graph with these properties simple. We call a sequence of edges e 1,...,e n a walk of length n if there is a corresponding sequence of vertices v 1,...,v n+1 such that e i = v i v i+1. A walk is called a 123

321 878 M. Keller, N. Peyerimhoff path if there is no repetition in the corresponding sequence of vertices v 1,...,v n. A (finite or infinite) path with associated vertex sequence...v i v i+1 v i+2...is called a geodesic,ifwe have d(v i,v j ) = i j for all pairs of vertices in the path. The boundary of a face f is the subgraph f = (V f, E f ).Wedefinethedegree f of a face f F to be the length of the shortest closed walk in the subgraph f meeting all its vertices. If there is no such finite walk, we set f =. Now we present the conditions that have to be satisfied in order that a planar graph be locally tessellating: Definition 2.1 A simple planar graph G is called a locally tessellating planar graph if the following conditions are satisfied: (i) Every edge is contained in precisely two different faces. (ii) Every two faces are either disjoint or have precisely a vertex or a path of edges in common. In the case that the length of the path is greater than one, then both faces are unbounded. (iii) Every face is homeomorphic to the closure of an open disc D R 2,toR 2 \D or to the upper half plane R R + R 2 and its boundary is a path. Note that these properties force the graph G to be connected. Examples are tessellations R 2 introduced in [2,3], trees in R 2, and particular finite tessellations on the sphere mapped to R 2 via stereographic projection. Now we turn to the proof of Theorem 1. The heart of the proof is Proposition 2.2 below. An earlier version of this proposition in the dual setting is Proposition 2.1 of [2]. We start with a few preliminary considerations. Let G = (V, E, F) be a locally tessellating planar graph. For a finite set W V let G W = (W, E W, F W ) be the subgraph of G induced by W, where E W are the edges in E with both end points in W and F W are the faces induced by the graph (W, E W ). Euler s formula states for a finite and connected subgraph G W (observe that F W contains also the unbounded face): W E W + F W =2. (9) Recall that E W is the set of edges connecting a vertex in W with one in V \W.By F W,we denote the set of faces in F which contain an edge of E W. Moreover, we define the inner degree of a face f F W by f i W = f W. We will need two useful formulas which hold for arbitrary finite and connected subgraphs G W = (W, E W, F W ). The first formula is easy to see and reads as v =2 E W + E W. (10) v W Since W is finite, the set F W contains at least one face which is not in F, namely the unbounded face surrounding G W, but there can be more. Define c(w ) as the number 123 c(w ) = F W F W F 1. (11)

322 Cheeger constants, growth and spectrum of locally tessellating planar graphs 879 Note that F W F is the number of faces in F which are entirely enclosed by edges of E W. Sorting the following sum over vertices according to faces gives the second formula 1 f = F W F + v W f v = F W c(w ) + f F W f i W f f F W f i W f. (12) Proposition 2.2 Let G = (V, E, F) be a locally tessellating planar graph and W Vbe a finite set of vertices such that the induced subgraph G W is connected. Then we have κ(w ) = 2 c(w ) E W 2 Proof By the Eqs. (9), (10)and(12) we conclude κ(w ) = 1 v f v W f v + f F W f i W f. = W E W E W 2 = 2 c(w ) E W 2 + F W c(w ) + + f F W f i W f. f F W f i W f A finite set W V is called a polygon,ifg W is connected and if c(w ) = 1. This notion becomes understandable if one looks at the dual setting: Every vertex v W corresponds to a face f (v) F in the dual planar graph G = (V, E, F ),andw V is a polygon if and only if v W f (v) R 2 is homeomorphic to a closed disc (here f denotes the closure of the geometric realization of the face f ). For v W,let v e W denote the number of edges in E W adjacent to v and call it the external degree of v (w.r.t. W ). Moreover, let V W be the set of vertices in W with v e W 1. Proposition 2.3 Ler G = (V, E, F) be a locally tessellating planar graph satisfying the vertex and face degree bounds in Theorem 1 and W V be a polygon with v e W p 2 for all v V W. Then we have E W 2C p,q (1 κ(w )). (13) Moreover, under the assumption of (a) or (b) in Theorem 1, wehave E W W 2C p,q C or E W vol(w ) 2C p,qc, respectively. Proof Observe first that we have the inequality f i W V W + E W. (14) f F W 123

323 880 M. Keller, N. Peyerimhoff This can be seen as follows: Every face f F W may have some edges and some isolated vertices in common with the induced graph G W = (W, E W, F W ). Since the vertices of V W are connected in G W, there are at least V W pairs ( f, e) F W E with e f E W. These pairs contribute at least 2 V W to the left hand sum in (14). At every vertex v V W, there are v e W 1 faces of F W which meet G W in the isolated vertex v. Adding over all these vertices v V W, we obtain the total contribution E W V W to the left hand sum in (14). One easily checks that there is no overlap of both contributions, leading to the above inequality. Using (14), V W p 2 1 E W, f q for all f F, and Proposition 2.2, we obtain ( 1 E W 2 p 1 ) q(p 2) E W 2 f F W f i W f = 1 κ(w ), which yields (13). The second formula of the proposition follows from κ(w ) C W in case (a) and from κ(w ) c vol(w ) in case (b). Henceforth, let G = (V, E, F) be a locally tessellating planar graph as in Theorem 1. Recall that C p,q =. The conditions v p and f q for all v V and q( p 2) (p 2)(q 2) 2 f F imply 2C p,q C p 2and2C p,q c p 2 p. Lemma 2.4 Let v V and W ={v}. Then we have E W = v 2C p,q C (15) W and E W vol(w ) = 1 p 2. p Proof The only non trivial inequality is (15). It follows in a straightforward way from κ(v) C that C q 2 2q v 1. This implies that (p 2)(q 2) 2C p,q C (p 2)(q 2) 2 v 2q(p 2) (p 2)(q 2) 2 2 v (q(p 2) p). (p 2)(q 2) 2 The lemma follows now from the fact that q(p 2) p 0forp, q 3. Lemma 2.5 Assume that there is a finite set W V such that E W E W < 2C p,q C p 2 or W vol(w ) < 2C p,qc p 2, respectively. (16) p Then there exists a polygon W V with v e W p 2 for all v V W, such that E W W E W W or E W vol(w ) E W vol(w ), respectively. 123

324 Cheeger constants, growth and spectrum of locally tessellating planar graphs 881 Proof Observe first that we can always find a non-empty subset W 0 W such that G W0 is a connected component of G W and that E W 0 / W 0 E W / W or E W 0 /vol(w 0 ) E W /vol(w ), respectively. Note that G W0 has only one unbounded face. By adding all vertices of V contained in the union of all bounded faces of G W0, we obtain a polygon with even smaller isoperimetric constants. Let us denote this non-empty polygon, again, by W. By Lemma 2.4, W must have at least two vertices. By connectedness of G W andbythe inequality W 2, we have v e W p 1forallv W.Assumethereisavertexv V W with v e W = p 1. Let W := W \{v}. Then one easily checks that the condition (16) implies or E W W = E W +2 p W 1 < E W W E W vol(w ) = E W +2 p < E W vol(w ) p vol(w ), respectively. Repeating this elimination of vertices with external degree p 1, we end up with a polygon W satisfying v e W p 2forallv W or with W equal to a single vertex. But the latter cannot happen by Lemma 2.4. Proof of Theorem 1 Since 2C p,q C p 2or2C p,q c p 2 p, we only have to consider the cases when α(g) <p 2or α(g) < p 2 p, since otherwise there is nothing to prove. Lemma 2.5 states that, if there is a finite W V with E W W < 2C p,q C or E W vol(w ) < 2C p,qc, respectively, then there is a polygon W with v e W p 2forallv V W satisfying the same inequality. But this contradicts Proposition 2.3, finishing the proof of Theorem 1. 3 Proofs of Theorems 2, 3 and 4 Let G = (V, E, F) be a q-face regular plane tessellation without cut locus, v 0 V, S n = S n (v 0 ) and σ n = S n. Recall that we have N = q 2 2 if q is even and N = q 2ifq is odd. The recursion formulas (5) in Theorem 2 for n N and n > N, respectively, require separate proofs. However, both proofs are based on the following results from [2, Sect. 6]. (Note that these results are presented there in the dual setting of vertex-regular graphs.) Proposition 6.3 in [2] states for n 1that κ(b n ) = 1 q 2 q 2 2q (σ n+1 σ n ) + where B n = {v V d(v 0,v) n} denotes the ball and j=2 q 2 j 2q c j n = { f F f (V \B n ) = j} c j n, (17) for 1 j q 1. Moreover from Lemma 6.2 in [2] we have the following recurrence relations for c j n, n 1, which arise very naturally from the geometric context (i) c l n = cl+2 n 1,for1 l q 3, (ii) c q 2 n = c 2 n 1, 123

325 882 M. Keller, N. Peyerimhoff (iii) cn q 1 = cn 1 + σ n+1 σ n = cn σ n+1 σ n. We first proof of (5)forn N. Letτ :{0, 1,...,N} {1, 2,...,q 1} be defined as q 1 2k if q is even and 0 k N, τ(k) = q 1 2k if q is odd and 0 k N 1 2, 2q 3 2k if q is odd and N+1 2 k N. Note that τ is defined precisely in such a way that we have c τ(k+1) n+1 = cn τ(k) for 0 k N 1andn 0, (18) by the recurrence relations (i) and (ii). Lemma 3.1 Let 1 n N. Then we have σ n+1 l σ n l for 1 l n 1, cn τ(l) = σ 1 for l = n, 0 for n + 1 l N. Moreover, in the case of even q, we have c 2l n = 0 for 1 l N. Proof One easily sees that c j 0 = 0for1 j q 2andcq 1 0 = σ 1. The recurrence relations (i) and (ii) imply that ck 1 = 0for0 k N 1andc1 N = σ 1. Using (iii), we obtain c q 1 k = σ k+1 σ k for 1 k N 1. The value of cn τ(l) can now be deduced from these results by repeatedly applying (18) in each of the cases 1 l n 1, l = n and n + 1 l N. Lemma 3.2 Let N 2, 1 n N and b l be defined as in (4). Then we have q 2 q 6 q 2 σ q 2 j n 1 n + q 2 c n j l=1 = b lσ n l if n N 1, 6 q j=2 q 2 σ 1 + N 2 l=1 b lσ N l if n = N. Proof First observe that, since 2 τ(l) q 2for1 l N 1andcn τ(l) n + 1 l N by Lemma 3.1 = 0for { q 2 nl=1 (q 2 j)cn j (q 2τ(l))cn τ(l) if n N 1, = (q 2τ(l))cτ(l) n if n = N, j=2 N 1 l=1 (Note for the case n = N:wehave2 τ(l) q 2 only for 1 l N 1andτ(N) = 1. This makes it necessary to treat this case separately.) The proof is now straightforward using the help of Lemma 3.1 and the equation (q 2)b l = 2(τ(l) τ(l + 1)). Proof of Theorem 2 We rewrite Eq. (17) as follows: σ n+1 σ n = 2q q 2 2q q 2 n κ(s l ) + l=0 Using κ(s 0 ) = 1 q 2 2q σ 1 and 2q q 2 κ(s l) = κ l σ l we obtain ( 123 σ n+1 = σ 1 ) n κ l σ l + σ n + l=1 q 2 j=2 q 2 j=2 q 2 j q 2 c j n. q 2 j q 2 c j n. (19)

326 Cheeger constants, growth and spectrum of locally tessellating planar graphs 883 The recursion formulas in Theorem 2 for N 2andn N follow now directly from (19) and Lemma 3.2. The case n = N = 1 has to be treated separately: In this case we have q 2 j=2 (q 2 j)c n j = 0and(19) simplifies to σ 2 = (2 κ 1 )σ 1. The result follows now from the fact that b 0 = 2. It remains to prove the recursion formula (5)forn > N. We first consider the case N 2. Repeated application of the recurrence relations (i) (iii) yields q 2 j=2 q 2 j q 2 c j n = ( N 1 l=0 b l σ n l ) (σ n + σ n (N 1) ) + q 2 j=2 q 2 j q 2 c j n N. (20) Since 2q q 2 (κ(b n) κ(b n N )) = N 1 l=0 κ n lσ n l, we obtain using (17)and(20) N 1 l=0 κ n l σ n l = (σ n+1 σ n ) + ( σ n (N 1) σ n N ) + ( N 1 l=0 b l σ n l ) ( σ n + σ n (N 1) ), which immediately yields the recursion formula (5) forn > N 2. The case N = 1is particularly easy and is left to the reader. This finishes the proof of Theorem 2. Now we turn to the proof of Theorem 3. Note first that non-positive vertex curvature implies non-positive corner curvature in the case of face-regular graphs. By [3, Theorem 1], both graphs G, G are without cut-loci and we can apply the recursion formulas in Theorem 2. Lemma 3.3 We have the following estimates for 0 l N 1 and k 1: b l κ k 2, if l = N 1 (= 0) and q {3, 4}, 2 b l κ k 1, if l = N 1 2 or q even, b l κ k 0, if l = N 1 (= 1) and q = 5, 2 b l κ k 1, if l = N 1 and q 7 odd. 2 Proof The case l = N 1 2 or q even follows from b l = q q 2 2q. Now assume that l = N 1 2.Sinceb l q 2 4 and κ(v) = 1 q 2 2q v 2, the previous considerations lead to b l κ k 1. If q = 3orq = 4, then b l = 2 and, consequently, b l κ k b l = 2. Finally, if q = 5, then κ(v) 0 implies that v 4 and thus κ(v) 1 4 q 2 2q. Using this fact leads directly to b l κ k 0. From the above lemma we deduce the following inequalities: Lemma 3.4 We have (a) b 0 κ k 1. (b) b N 1 κ k 1 if N 2. (c) b 0 κ k 2 if q = 3 or q =

327 884 M. Keller, N. Peyerimhoff (d) Let n, N 1, 1 k min{n, N}, and assume that γ i 0 are mononote non-decreasing for n k + 1 i n 1.Then k 1 (b l κ n l )γ n l 0. l=1 Proof (a), (b) and (c) are trivial consequences of Lemma 3.3. (d) follows immediately from Lemma 3.3, unless we have k 1 N 1 2 and q 7 odd. But in this case we have N and k 1 (b l κ n l )γ n l (b 1 κ n 1 )γ n 1 + (b N 1 2 l=1 by the monotonicity of γ i. γ n 1 γ n N 1 2 0, κ n N 1 )γ 2 n N 1 2 Proof of Theorem 3 Let κ n = κ( S n ). Then the condition κ( S n ) κ(s n ) 0 reads κ n κ n for n 0. Note that σ 0 = σ 0 = 1. Since σ 1 = 2q q 2 κ 0, σ 1 = 2q q 2 κ 0 we conclude that σ 1 σ 1 = κ 0 κ 0 0 = σ 0 σ 0. The proof is based on induction over n: Assume that n 1andthatγ k := σ k σ k is non-negative and monotone non-decreasing for 0 k n. We aim to show that γ n+1 γ n. We first consider the case n N. Then the recursion formula (5) yields n 1 γ n+1 γ n + (b l κ n l )γ n l, (21) l=1 and γ n+1 γ n follows from Lemma 3.4(d). Finally, we consider the case n > N.IfN 2, the recursion formula (5), Lemma 3.4(a,b) and the monotonicity of γ k yields γ n+1 γ n + ( N 1 ) (b l κ n l )γ n l γ n N l=1 N 2 γ n + (b l κ n l )γ n l. l=1 Again, γ n+1 γ n follows now from Lemma 3.4(d). If N = 1 (i.e., q = 3orq = 4), the recursion formula (5) simplifies considerably and, using Lemma 3.4(c), we conclude that γ n+1 2γ n γ n 1 γ n, finishing the proof of Theorem 3. Proof of Theorem 4 Let v 0 V be arbitrary and S n, n N 0 the corresponding distance spheres. Since we assumed G to be connected and the vertex degree to be uniformly bounded from above we have μ(g) = μ(g,v 0 ) = μ(g,v 0 ) = μ(g). LetT be a spanning tree of G which leaves the distance relation with respect to v 0 invariant. This can easily be achieved as follows: One removes all edges which connect vertices within a sphere. Further one removes inductively all edges which connect a vertex in S n to vertices in S n 1 except for one edge. 123

328 Cheeger constants, growth and spectrum of locally tessellating planar graphs 885 Since the sphere structure is left invariant the spanning tree T has the same exponential volume growth as G, i.e., μ(g,v 0 ) = μ(t,v 0 ). Moreover T can be embedded into a p-regular tree T p since the vertex degree in T is smaller or equal than the supremum of the vertex degrees in G which we denoted by p. This finishes the proof. Acknowledgments Matthias Keller likes to thank Daniel Lenz who encouraged him to study the connection between curvature and spectral theory. Matthias Keller was financially supported during this work by SDW. Norbert Peyerimhoff is grateful for the financial support of the Technical University of Chemnitz. Both authors like to thank Ruth Kellerhals and Victor Abrashkin for very useful discussions. Moreover the authors very much appreciate the comments of Radosław Wojciechowski which let to a simplification of the proof of Theorem 4. References 1. Bartholdi, L., Ceccherini-Silberstein, T.G.: Salem numbers and growth series of some hyperbolic graphs. Geom. Dedicata 90, (2002) 2. Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25(1), (2001) 3. Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), (2006) 4. Cannon, J.W., Wagreich, P.: Growth functions on surface groups. Math. Ann. 293(2), (1992) 5. DeVos, M., Mohar, B.: An analogue of the Descartes-Euler formula for infinite graphs and Higuchi s conjecture. Trans. Am. Math. Soc. 359(7), (2007) 6. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am.Math.Soc.284(2), (1984) 7. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Durrett, R., Pinsky, M.A. (eds.) Geometry of Random Motion, vol. 73, pp AMS Contemporary Mathematics (1988) 8. Dodziuk, J., Kendall, W.S. : Combinatorial Laplacians and isoperimetric inequality. In: Elworthy, K.D. (ed.) From Local Times to Global Geometry, Control and Physics, pp Longman Scientific and Technical, Harlow (1986) 9. Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46(3), (1979) 10. Floyd, W.J., Plotnick, S.P.: Growth functions on Fuchsian groups and the Euler characteristic. Invent. Math. 88(1), 1 29 (1987) 11. Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. 48(2), (1996) 12. Fujiwara, K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), (1996) 13. Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 2nd edn, Universitext. Springer-Verlag, Berlin (1990) 14. Gromov, M.: Hyperbolic Groups. Essays in Group Theory, pp Math. Sci. Res. Inst. Publ. 8. Springer, New York (1987) 15. Häggström, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30(1), (2002) 16. Higuchi, Y.: Combinatorial curvature for planar graphs. J. Graph Theory 38(4), (2001) 17. Higuchi, Y.: Boundary area growth and the spectrum of discrete Laplacian. Ann. Glob. Anal. Geom. 24(3), (2003) 18. Higuchi, Y., Shirai, T.: Isoperimetric constants of (d, f )-regular planar graphs. Interdiscip. Inform. Sci. 9(2), (2003) 19. Higuchi, Y., Shirai, T.: Some spectral and geometric properties for infinite graphs. In: Discrete Geometric Analysis, pp Contemp. Math Amer. Math. Soc., Providence (2004) 20. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43(4), (2006) 21. Keller, M.: The essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), (2010) 22. Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. Preprint 23. Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. In: Mathematical Modelling of Natural Phenomena, Spectral Problems (to appear) 123

329 886 M. Keller, N. Peyerimhoff 24. Klassert, S., Lenz, D., Peyerimhoff, N., Stollmann, P.: Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proc. Am. Math. Soc. 134, (2005) 25. McKean, H.P.: An upper bound to the spectrum of on a manifold of negative curvature. J. Differ. Geom. 4, (1970) 26. Mohar, B.: Light structures in infinite planar graphs without the strong isoperimetric property. Trans. Am.Math.Soc.354(8), (2002) 27. Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21(3), (1989) 28. Stone, D.A.: A combinatorial analogue of a theorem of Myers and Correction to my paper: A combinatorial analogue of a theorem of Myers. Illinois J. Math. 20(1), (1976); 20(3), (1976) 29. Urakawa, H.: The spectrum of an infinite graph. Can. J. Math. 52(5), (2000) 30. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. abs/ Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124(3), (1998) 32. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics, vol Cambridge University Press, Cambridge (2000) 33. Wojciechowski, R.K.: Stochastic completeness of graphs. PhD thesis (2007) 34. Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), (2009) 123

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332 Discrete Comput Geom (2011) 46: DOI /s Curvature, Geometry and Spectral Properties of Planar Graphs Matthias Keller Received: 4 May 2010 / Revised: 15 January 2011 / Accepted: 9 February 2011 / Published online: 1 March 2011 Springer Science+Business Media, LLC 2011 Abstract We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger s constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of the essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies the absence of finitely supported eigenfunctions for nearest neighbor operators. Keywords Discrete curvature Hyperbolic properties Cut locus Cheeger s constant Exponential growth Spectrum of graphs Unique continuation of eigenfunctions 1 Introduction There is a long tradition of studying planar graphs, i.e., graphs that can be embedded in a topological surface homeomorphic to R 2 without self intersection. In particular, the curvature of tessellating graphs received a great deal of attention during the recent years, see for instance [2 4, 9, 11, 15, 19, 20, 25 27, 30] and references therein. Here, we introduce a notion of curvature for general planar graphs to study their geometry M. Keller ( ) Mathematical Institute, Friedrich-Schiller-University Jena, Jena, Germany m.keller@uni-jena.de

333 Discrete Comput Geom (2011) 46: and spectral properties. In the case of tessellations, which are also called tilings, the definitions coincide with the one of [2, 3, 15, 25, 27]. We next summarize and discuss the main results of the paper. Precise formulations can be found in the sections, where the results are proven. The key insight of this paper is that non-positive curvature alone has already very strong implications on the structure of a planar graph, i.e.: (1) The graph is infinite and locally tessellating (Theorem 1 in Sect. 3). (2) The graph is locally similar to a tessellation (Theorem 2 in Sect. 4). Let us comment on these points. In contrast to the statement about the infinity, positive curvature implies finiteness of the graph. This was studied in [4, 9, 15, 25, 26]. Note also that locally tessellating graphs, recently introduced in [20] (see also [30]), allow for a unified treatment of planar tessellations and trees. The idea of (2) is to construct an embedding of a locally tessellating graph into a tessellation that preserves crucial properties of a fixed finite subset. This embedding allows us to carry over many results known for tessellations to planar graphs. Indeed, most of the results are then direct corollaries of (1), (2) and the corresponding results for tessellations. In particular, we obtain for planar graphs with non-positive curvature: (3) Absence of cut locus, i.e., every distance minimizing path can be continued to infinity (Theorem 3 in Sect. 5.1). (4) A description of the boundary of distance balls (Theorem 4 in Sect. 5.2). For regular tessellations a result similar to (3) was obtained by Baues, Peyerimhoff in [2] and was later extended to general tessellations by the same authors in [3]. This result is considered a discrete analog of the Hadamard Cartan theorem in differential geometry. Questions about the possible depth of the cut locus for Cayley graphs were studied under the name dead-end-depth in [5], see also references therein. The boundary structure of distance balls, mentioned in (4), plays a crucial role in the techniques of Żuk [30]. These concepts were later refined by [2, 3] under the name admissibility of distance balls. Here, we generalize the most important statements of [3, 30] to planar graphs. For negatively curved planar graphs, we prove the following: (5) Bounds for the growth of distance balls (Theorem 5 in Sect. 5.3). (6) Positivity and bounds for Cheeger s constant (Theorem 6 in Sect. 5.4). (7) Empty interior for minimal bigons (Theorem 7 in Sect. 5.5). In [2], a result similar to (5) is shown for tessellations. As for (6), positivity of Cheeger s constant was proven by Woess [27] for tessellations and simultaneously by Żuk in a slightly more general context. Later, this was rediscovered in [15]. More precisely, Żuk showed that the norm of the transition matrix is strictly smaller than one. However, this is equivalent to positivity of Cheeger s constant, see [7, 8, 14]. Here, we give new explicit bounds for Cheeger s constant, which can be interpreted in terms of the curvature. For related results, see for instance [16, 17, 19, 20]. The results (6) and (7) can be understood as hyperbolic properties of a graph. In particular, if G is the Cayley graph of a finitely generated group both results imply Gromov hyperbolicity. In general, this is not the case in our situation. However, under the additional assumption that the number of edges of finite polygons is uniformly bounded,

334 502 Discrete Comput Geom (2011) 46: empty interior for minimal bigons, along with Gromov hyperbolicity can be proven, see [3, 30]. Let us now turn to the spectral implications. It is well known that bounds for Cheeger s constant imply bounds for the bottom of the spectrum of the graph Laplacian, see [1, 6, 8, 12, 22]. The Cheeger constant also plays an important role in random walks. In particular, the simple random walk is transient if Cheeger s constant is positive, see [14, 27, 28] and also [24]. Along with these well known relations, we show for planar graphs with non-positive curvature: (8) Triviality of the essential spectrum of the Laplacian if and only if the curvature decreases uniformly to (Theorem 8 in Sect. 6.1). (9) Absence of finitely supported eigenfunctions for nearest neighbor operators (Theorem 9 in Sect. 6.2). Statement (8) generalizes results of [19] for tessellations and of [12] for trees. Statement (9) is an extension of [21]. As for general planar graphs eigenfunctions of finite support can occur, this unique continuation statement shows that the analogy between Riemannian manifolds and graphs is much stronger in the case of non-positive curvature. The paper is structured as follows. In Sect. 2, we give the basic definitions. Section 3 is devoted to the statement and the proof of Theorem 1. In Sect. 4, we construct the embedding into tessellations. Sect. 5 concentrates on the geometric applications (3) (7) and Sect. 6 is devoted to spectral applications (8) and (9). 2 Definitions and a Combinatorial Gauss Bonnet Formula Let G = (V, E) be a graph embedded in an oriented topological surface S. Anembedding is a continuous one-to-one mapping from a topological realization of G into S. A graph admitting such an embedding into R 2 is referred to as planar. We will identify G with its image in S. The faces F of G are defined as the closures of the connected components of S \ E. We write G = (V,E,F). We call G simple if it has no loops (i.e., no edge contains only one vertex) and no multiple edges (i.e., two vertices are not connected by more than one edge). We say that G is locally finite if for every point in S there exists an open neighborhood of this point that intersects with only finitely many edges. A characterization whether a Cayley graph of a group allows for a planar embedding that is locally finite was recently given in [13]. For the rest of this paper, we will assume that G is locally finite. We call two distinct vertices v,w V adjacent and we write v w if there is an edge containing both of them. We also say that two elements of V, E and F (possibly of distinct type) are adjacent if their intersection is non-empty and we call two adjacent elements of the same type neighbors. By the embedding, each edge corresponds uniquely to a curve (up to parametrization). We call a sequence of edges a walk if the corresponding subsequent curves can be composed to a curve. Here, we allow for two sided infinite sequences. The length of the walk is the number of elements of the sequence, whenever there are only finitely many and infinite otherwise. We refer to a subgraph, where the edges can be

335 Discrete Comput Geom (2011) 46: ordered to form a walk and each vertex is contained in exactly two edges as a path. The length of a path is the length of the shortest walk passing all vertices. We call vertices that are contained in two edges of a walk or a path the inner vertices of the walk or the path. A walk or a path is called closed if every vertex contained in it is an inner vertex and simply closed if all vertices are contained in exactly two edges. Note that every simply closed walk induces a path and every closed path is simply closed. We say a graph is connected if any two vertices can be joined by a path. For the remainder of this paper, we will assume that G is connected. For a face f F, we call a walk a boundary walk of f if it meets all vertices in f and if it is closed whenever it is finite. The existence of boundary walks for all faces is ensured by the connectedness of the graph. We let the degree f of a face f F be the length of the shortest boundary walk of f, whenever f admits a finite boundary walk and infinite otherwise. We define the degree e of an edge e E as the number of vertices contained in e. For a vertex v V, we define the degree by v :=2 e E,v e 1 e. The formula can be interpreted as the number of adjacent edges with degree two plus twice the number of adjacent edges with degree one. The latter are counted twice since they meet v twice. A vertex v V with v =1 is called a terminal vertex. To define curvature functions, we first have to introduce the corners of a planar graph G.Thesetofcorners C(G) is a subset of V F such that the elements (v, f ) satisfy v f. We denote for a vertex v the set C v (G) := {(v, g) C(G)} and for a face f the set C f (G) := {(w, f ) C(G)}. Thedegree or the multiplicity (v, f ) of a corner (v, f ) C(G) is the minimal number of times the vertex v is met by a boundary path of f. For tessellations, the definition of corners coincides with the one in [2, 3] and there every corner has degree one. However, if G is not a tessellation the degree of a corner can be larger than one (see for instance the left hand side of Fig. 1 in the next section). For a vertex v V and a face f F,wehave v = (v,g) C v (G) (v, g) and f = (w,f ) C f (G) (w, f ). We can think of the corners of a vertex v with respect to the multiplicity as the partitions of a sufficiently small ball after removing the edges adjacent to v (where small means that no edge is completely included in the ball and there are exactly v partitions). The curvature function on the corners κc G : C(G) R is defined by κ G C (v, f ) := 1 v f, with the convention that 1/ f =0 whenever f =. We define the curvature function on the vertices κ G V : V R by κ G V (v) := (v,f ) C v (G) (v, f ) κ G C (v, f ).

336 504 Discrete Comput Geom (2011) 46: By direct calculation, we arrive at κ G V (v) = 1 v 2 + (v,f ) C v (G) (v, f ) 1 f. We define the curvature function on the faces κf G : F [, ) by (f ) := (v, f ) κ G C (v, f ). κ G F (v,f ) C f (G) If f = and if there are infinitely many vertices in f with vertex degree of at least three, then the curvature of a face f takes the value. For finite subsets V V, F F, we write κv G (V ) := κv G (v) and κg F (F ) := κf G (f ). v V f F Moreover, we let κ C (G) := sup κc G (v, f ), κ (v,f ) C(G) V (G) := sup κv G (v), v V κ F (G) := sup κf G (f ). f F We call a face a polygon if it is homeomorphic to the closure of the unit disc D in R 2 and its boundary is a closed path. We call a face an infinigon if it is homeomorphic to R 2 \ D or the upper half-plane R R + R 2 and its boundary is a path. A graph G is called tessellating if the following conditions are satisfied: (T1) Every edge is contained in precisely two different faces. (T2) Every two faces are either disjoint or intersect precisely in a vertex or in an edge. (T3) Every face is a polygon. The additional assumption in [2, 3] that each vertex has finite degree is already implied by local finiteness of G. Note that the dual graph of a non-positively corner curved tessellation is also a non-positively corner curved tessellation. For a discussion, we refer to [2]. We call a path of at least two edges an extended edge if all inner vertices have degree two and the beginning and ending vertex, in the case that they exist, have degree greater than two. An extended edge contained in two faces with infinite degree is called regular. We introduce two weaker conditions than (T2) and (T3): (T2 ) Every two faces are either disjoint or intersect precisely in a vertex, an edge or a regular extended edge. (T3 ) Every face is a polygon or an infinigon. We call a graph satisfying (T1), (T2), (T3 ) strictly locally tessellating and a graph satisfying (T1), (T2 ), (T3 ) locally tessellating. Note that a locally tessellating graph that contains no extended edge is strictly locally tessellating. While the graphs studied in [2 4, 9, 15, 17, 19, 21, 24 27] are tessellating, the results of [20, 30] concern strictly locally tessellating graphs. We give some examples.

337 Discrete Comput Geom (2011) 46: Examples (1) Tessellations of the plane R 2 or the sphere S 2 are strictly locally tessellating. Furthermore, tessellations of S 2 embedded into R 2 via stereographic projection are also strictly locally tessellating. (2) Trees are strictly locally tessellating if and only if the branching number of every vertex is greater than one. In this case, the tree is negatively curved in each corner, vertex and face. (3) The graph with vertex set Z and edge set {[n, n + 1]} n Z is locally tessellating, but not strictly locally tessellating and has curvature zero in every corner, vertex and face. Next, we will prove a combinatorial Gauss Bonnet formula. We refer to [2] for background and proof in the case of tessellations, see also [4, 9] for further reference. Let G = (V,E,F) be a planar graph embedded into S 2 or R 2. For a subset W V, we denote by G W = (W, E W,F W ) the subgraph of G induced by the vertex set W, where E W E are the edges that contain only vertices in W and F W are the faces induced by the graph (W, E W ). For a finite connected subset of vertices W V, Euler s formula reads V W E W + F W =2. Observe that F W contains also an unbounded face, which explains the two on the right hand side. Proposition 1 (Gauss Bonnet formula) Let G be planar and W V finite and connected. Then Proof We have, by definition, κ G W V (W ) = (v,f ) C(G W ) = v W c C v (G W ) κ G W V (W ) = 2. (v, f ) ( 1 v + 1 f 1 ) 2 c v + f F W c C f (G W ) c f v W c C v (G W ) Since v = c C v (G W ) c, the first term is equal to W.AsW is connected, we have f = c C f (G W ) c. Thus, the second term is equal to F W. LetE v,j ={e E W v e, e =j} for v W and j = 1, 2. Then, c C v (G W ) c = v =2 E v,1 + E v,2 by the definition of v. Moreover, for each e E v,2 there is a unique w W, w v, such that e E w,2. We conclude v W c C v (G W ) c 2 = E v, E v,2 = E W. v W c 2. By Euler s formula, we obtain the result.

338 506 Discrete Comput Geom (2011) 46: Remark The Gauss Bonnet formula immediately implies that a finite graph must admit some positive curvature. This is, in particular, the case for locally finite planar graphs embedded in S 2. We write d(v,w) for the length of the shortest path connecting the vertices v and w. For a set of vertices W V, we define the balls and the spheres of radius n by B n (W ) := B G n (W ) := { v V d(v,w) n for some w W }, S n (W ) := S G n (W ) := B n(w ) \ B n 1 (W ). We define the boundary faces of W by F W := G F W := { f F f W,f V \ W }. 3 Infinity and Local Tessellating Properties of Non-positively Curved Planar Graphs This section is dedicated to prove the first main result. Theorem 1 Let G be a planar graph that is connected and locally finite. If one of the following conditions is satisfied (a) κ C (G) 0 or (b) κ V (G) 0 and G is simple or (c) κ F (G) 0, each extended edge is regular and there are no terminal vertices, then G is infinite and locally tessellating. If a strict inequality holds in condition (a) or (b) then G is even strictly locally tessellating. The main idea of the proof can be summarized as follows. The infinity is a direct consequence of the Gauss Bonnet formula. Moreover, assuming a graph is not locally tessellating, we will construct a finite planar graph with smaller or equal curvature, which can be embedded into S 2. By the Gauss Bonnet formula, this graph must admit some positive curvature. As the curvature is smaller or equal, the original graph must have some positive curvature as well. We start with analyzing the pathologies that occur in general planar graphs that are not locally tessellating. Let G = (V,F,E) be a planar graph that is connected and locally finite. A face f F is called degenerate if it contains a vertex v such that (v, f ) 2 for the corner (v, f ). Note that this is in particular the case if v is contained in three or more boundary edges of f or there is an edge that is included in no other face except for f. A pair of faces (f, g) is called degenerate if f g consists of at least two connected components. Figure 1 shows examples of degenerate faces. We denote by D(F) F the set of all faces that are degenerate or are contained in a degenerate pair of faces.

339 Discrete Comput Geom (2011) 46: Fig. 1 Examples of a degenerate face f and a degenerate pair of faces (f, g) Fig. 2 The figure shows how to isolate a finite subset W, whenever there is degenerate pair of faces or a degenerate face Lemma 1 Let G = (V,E,F) beasimple, planar graph that is connected, locally finite and contains no terminal vertices. If D(F), then there is a finite and connected subgraph G 1 = (V 1,E 1 ), which is bounded by a simply closed path and satisfies the following property: there are at most two vertices v 1,v 2 V 1 such that each path connecting V 1 and V \ V 1 meets either v 1 or v 2. In particular, v 1, v 2 lie in the boundary path of G 1. Proof The proof consists of two steps. Firstly, we find a finite set W V that contains at most two vertices with the asserted property. We have to deal with the case of a degenerate face and a degenerate pair separately. Secondly, we find a subgraph of G W that has a simply closed boundary. We start with the case of a degenerate pair of faces (f, g). Letw 1,w 2 be two vertices that are contained in different connected components of the intersection of f and g. Letγ f be a simple curve that lies in (int f) {w 1,w 2 } and connects w 1 and w 2, where int f := f \ {e E e f }. Similarly, let γ g be a simple curve in (int g) {w 1,w 2 } connecting w 1 and w 2. Composing these curves, we obtain a simply closed curve γ which divides the plane by Jordan s curve theorem into a bounded and an unbounded component. We denote the bounded component by B.

340 508 Discrete Comput Geom (2011) 46: Fig. 3 The right hand side shows an enumeration of the boundary walk around G W illustrated on the left hand side. To isolate the subgraph G 1, we pick the simply closed subwalk from w 3 to w 3 The set W = V B can be connected to V \ W only by walks which meet w 1 or w 2. For an illustration see the left hand side of Fig. 2. We now turn to the case when there is no degenerate pair of faces. As D(F), there must be a degenerate face f.letw 1 be a vertex in f with (w 1,f) 2. We pick an open simply connected neighborhood U of w 1 that contains no other vertex except for w 1 and e U is connected for all edges e adjacent to w 1.Letx 1 and x 2 be two arbitrary points in two different connected components of (int f) U. We connect x 1 and x 2 by a simple curve in int f and connect x 1,x 2 and w 1 also by simple curves that lie in the corresponding connected component of U. By composition, we obtain a simply closed curved and by Jordan s curve theorem we get a bounded set B. Note that W = V B can be connected to V \ W only by walks which meet w 1.For an illustration see the right hand side of Fig. 2. In both cases, we identified a set W which is finite and connected since the graph G is locally finite and connected. Now, we find a subset V 1 W such that G 1 = (V 1,E 1 ) is bounded by a simply closed path. By construction, w 1 lies in an unbounded face of F W, which we denote by f and which has a finite boundary walk. We start from w 1 walking around f. The walk might visit certain vertices several times. The vertex w 1 is visited at least twice, since the walk is finite. We pick a subsequence of the walk such that no vertex is met twice except for the starting and ending vertex, which we denote by v 1.InFig.3, this is illustrated (with v 1 = w 3 ). The sequence of edges forms a closed path and encloses a subgraph, which we denote by G 1 = (V 1,E 1 ). Note that V 1 V contains v 1, which might be equal to w 1 and V 1 also might contain w 2, which we denote in this case by v 2. Nevertheless, v 1 and v 2 are the only vertices in V 1 that can be connected to V \ V 1 by edges of G. Thus, G 1 has the desired properties and we finished the proof. The next lemma is the main tool for the proof of Theorem 1. It shows that the existence of degenerate faces or pairs implies the presence of positive curvature. Lemma 2 (Copy and paste lemma) Let G = (V,E,F) be a simple, planar graph that is connected, locally finite, contains no terminal vertices and all extended edges are regular. If D(F), then there are subsets F F and V V such that κ G F (F )>0 and κ G V (V )>0. Proof The idea of the proof is easy to illustrate. We assume D(F) and take the subgraph G 1 of Lemma 1. We make copies of G 1, paste them along the boundary

341 Discrete Comput Geom (2011) 46: Fig. 4 An illustration of the copy and paste procedure and embed the resulting graph into S 2. See Fig. 4. We then show that the curvature of this graph compared to the curvature in G does not increase, as long as we make enough copies of G 1. The statement is then implied by the Gauss Bonnet formula. Assume D(F). By Lemma 1, there is a finite subgraph G 1 = (V 1,E 1,F 1 ) of G which is enclosed by a closed path p. The vertex degree in G 1 differs from G in at most two vertices, which we denote by v 1 and v 2. (If there is only one vertex, then we choose another vertex arbitrarily in the boundary path of G 1.) Let {e 1,...,e n } be the edges of the boundary path of G 1 starting and ending at v 1, i.e., v 1 = e 1 e n. Moreover, let m<n such that v 2 = e m e m+1. Denote p 1 ={e 1,...,e m } and p 2 ={e m+1,...,e n }. We take two copies G (1) 1 and G (2) 1 of G 1. We paste them