SEMICLASSICAL LIMITS OF QUANTUM PARTITION FUNCTIONS ON INFINITE GRAPHS

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1 SEMICLASSICAL LIMITS OF QUANTUM PARTITION FUNCTIONS ON INFINITE GRAPHS BATU GÜNEYSU Abstract. We prove that if H denotes the operator corresponding to the canonical Dirichlet form on a possibly locally infinite weighted graph X, b, m), and if v : X R is such that H + v/ is well-defined as a form sum for all > 0, then the quantum partition function tre H+v/ ) ) satisfies tre H+v/ ) ) 0+ e βvx) for all β > 0, regardless of the fact whether e βv is apriori summable or not. This allows geometric Weyl-type convergence results. We also prove natural generalizations of this semiclassical limit to a large class of covariant Schrödinger operators that act on sections in Hermitian vector bundle over X, m, b), a result that particularly applies to magnetic Schrödinger operators that are defined on X, m, b). 1. Introduction Let us recall some classical facts from the Euclidean R l : Assume that + v is a Schrödinger operator in L 2 R l ) with v : R l R say) bounded from below and e βv L 1 R l ) for some β > 0. Then one has the following behaviour of the corresponding quantum partition function tre β 2 +v ): 2π) l l 2 tr e β +v)) 0+ p 2 1) 2 2 )dpdq. +vq) R l R l e β Clearly, 1) is a semiclassical limit, as the integral on the right hand side is over the classical phase space of the system, and thus can be seen as the classical partition function of the system. The analogous result holds true with the same right hand side as in 1), if one couples to a magnetic field. If one thinks about realizing analogous results in a discrete configuration space, one is apriori faced with the problem that it is not really Date: October 13,

2 2 BATU GÜNEYSU clear what the underlying phase space should be. However, if one realizes that 1) is equivalent to 2π ) l 2 tr e 1 +v/ )) 0+ 2) 2 e βvx) dx, R l it is clear that the latter problem only exists conceptually, not mathematically. A typical path integral proof [20] of 2) relies on a Golden- Thompson type inequality of the form tr e t 1 +v)) 3) 2 e t 2 x, x)e tvx) dx for all t > 0, R l which is of course = 2πt) l 2 e tvx) dx R l and thus shows that the necessity of the regularizing factor 2π) l 2 in 2) ultimately comes from the on-diagonal singularity at t = 0 of e t 2, ). On the other hand, it has been shown recently [6, 7] that if H is the generally unbounded) operator corresponding to the canonical Dirichlet form on an arbitrary possibly locally infinite weighted graph 1 X, b, m) and if v : X R is a potential such that H +v is well-defined as a form sum, then one still has a Golden-Thompson inequality 4) tr e th+v)) e th x, x)e tvx) mx) for all t > 0, the fundamental difference of this discrete setting to the continuum setting is, however, that one has e th x, y)mx) 1, as e th x, y)mx) is precisely the probability of finding the underlying Markoff particle in y at the time t, when conditioned to start in x. In particular, the right hand side of 4) is e tv. At this point, at the latest, it becomes natural to ask whether the quantum partition function tr e H+v/ )) satisfies 5) tr e H+v/ )) 0+ e βvx). One of our main results, Theorem 2.8, states that 5) indeed is always true, whenever v is such that H +v/ is well-defined as a form sum for all > 0. Here, in contrast to the continuum setting, we even do not 1 Here, X is a countable set of vertices, b : X X [0, ) an edge weight function, and m : X 0, ) a vertex weight function; cf. section 2.1 for the precise definitions.

3 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 3 have to assume e βv <, in the sense that the limit in 5) exists anyway. This admits entirely geometric results such as 6) tr e H+ m/ )) 0+ mx), with m := lnm) : X R, under very weak assumptions on m. Finally, using results from [7], we extend Theorem 2.8 to covariant Schrödinger operators that act on sections in Hermitian vector bundles over X, b, m) in Theorem Firstly, in view of the corresponding Feynman-Kac formulae cf. [5, 7]), these discrete operators appear as approximations to covariant Schrödinger semigroups on manifolds, as [11] one can approximate Brownian motion by time-continuous geodesic random walks on any possibly noncompact Riemannian manifold cf. [1], where this operator approximation has been worked out in detail in flat space). Secondly, these discrete covariant Schrödinger operators have found direct applications in combinatorics in generalized matrixtree theorems), physics and image processing [21, 12, 2]. In particular, Theorem 2.10 applies to magnetic Schrödinger operators which have been considered in [22, 18, 16, 17, 15, 4, 14], and more recently in [6]. This perfectly matches with the magnetic variant of 1). Our proofs of Theorem 2.8 and Theorem 2.10 are entirely probabilistic, and they rely on various covariant) Feynman-Kac formulae that have been established recently in [6, 7]. This paper is organized as follows: Section 2 is devoted to the formulation of the main results, Theorem 2.8 and Theorem To this end, we first explain the underlying Dirichlet space setting of weighted discrete graphs and Hermitian vector bundles thereon in Section 2.1. In Section 2.2, we give precise functional analytic definitions of the underlying covariant) Schrödinger operators, so that finally Theorem 2.8 and Theorem 2.10 can be formulated in Section 2.3. Then Section 3 is devoted to the proofs of these two theorems. Since, as we have already remarked, these proofs use partially technical probabilistic results from [6, 7], we first completely develop the corresponding notation in Section 3 for the convenience of the reader. Acknowledgements. The author B.G.) would like to thank Francoise Truc and Matthias Keller for helpful discussions. B.G. has been financially supported by the SFB 647: Raum-Zeit-Materie. 2. Main results

4 4 BATU GÜNEYSU 2.1. Setting. Let X, b, m) be an arbitrary weighted graph, that is, X is a countable set, b is a symmetric function with the property that bx, x) = 0, b : X X [0, ) bx, y) < for all x X, and m : X 0, ) is an arbitrary function. Here, X is interpreted as the set of vertices, {b > 0} is interpreted as the set of weighted) edges of the graph X, b), and m is considered as a vertex weight function see also Remark 2.4 below). Given x, y X we write x b y, if and only if bx, y) > 0, and X, b) is called connected, if for all x, y X there is a finite chain x 1,..., x n X such that x = x 1, y = x n and x j b x j+1 for all j = 1,..., n 1. As m determines a measure on the discrete space X in the obvious way, we get the corresponding complex Hilbert space of complex-valued m- square-summable functions on X, which will be denoted by l 2 mx). We use the conventions deg m,b x) := 1 bx, y), deg mx) 1,b x) := bx, y) for all x X, and the spaces of complex-valued and complex-valued finitely supported functions on X will be denoted by CX) and C c X), respectively, where of course C c X) is dense in l 2 mx). We fix an arbitrary w.l.o.g.) connected weighted graph X, b, m). Next, we recall that a complex vector bundle F X on X with rankf ) = ν N is given by a family F = F x of ν-dimensional complex linear spaces, where then the corresponding space of sections is denoted by ΓX, F ) = { f f : X F, fx) Fx }. If additionally each fiber F x comes equipped with a complex scalar product, ) x, then F X is referred to as a Hermitian vector bundle, and the norm and operator norm corresponding to, ) x will be denoted with x. We get the corresponding complex Hilbert space of l 2 m-sections Γ l 2 m X, F ) = { f f ΓX, F ), f l 2 m X) } with its canonical scalar product f 1, f 2 m := x xf 1 x), f 2 x)) x mx)

5 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 5 and norm f m := f, f m, where of course m will also denote the corresponding operator norm. Note that Γ l 2 m X, F ) generalizes l 2 mx) in a canonic way cf. Remark 2.2 below), and that the space of finitely supported sections Γ Cc X, F ) is dense in Γ l 2 m X, F ). For any x X we write 1 x : F x F x for the identity operator and tr x : EndF x ) C for the underlying canonical trace. In the above situation, a unitary b-connection Φ on F X is an assignment which to any x b y assigns an unitary map Φ x,y : F x F y with Φ y,x = Φ 1 x,y. We fix a Hermitian vector bundle F X of rank ν N, with a unitary b-connection Φ defined on it Operators under consideration. We start by defining a sesquilinear form Q Φ,0 in Γ l 2 m X, F ) with domain of definition Γ c X, F ) by setting Q Φ,0 f 1, f 2 ) = 1 bx, y) f 1 x) Φ y,x f 1 y), f 2 x) Φ y,x f 2 y) ) 2. x x b y The form Q Φ,0 is densely defined, symmetric and nonnegative. Furthermore, one has: Lemma 2.1. QΦ,0 is closable. Proof. It is sufficient to show that the map Γ l 2 m X, E) [0, ], f 1 bx, y) fx) Φy,x fy) 2 2 x x b y is lower semicontinuous. However, from the discreteness of the underlying measure space we get the implication n n f n f m 0 f n z) fz) z 0 for all z X, so that the claim follows from Fatou s lemma. We denote the closure of QΦ,0 by Q Φ,0 and the self-adjoint operator corresponding to the latter form by H Φ,0. Remark 2.2. Of course, we can deal with usual scalar functions on X simply by taking F x = {x} C with its canonical Hermitian structure. Indeed, then the sections in F X can be canonically identified with complex-valued functions on X, and any Φ can uniquely be written as Φx, y) = e iθx,y) with θ is a magnetic potential on X, b), that is, an antisymmetric function θ : {b > 0} [ π, π], and as a particular case scal of the above constructions, we get the corresponding forms Q θ,0 and Q scal θ,0 in l2 mx). The operator corresponding to Q scal θ,0 will be denoted with Hθ,0 scal. The particular form Q := Qscal 0,0 is a regular Dirichlet form

6 6 BATU GÜNEYSU l 2 mx), when X is equipped with its discrete topology. Let H := H scal 0,0 denote the corresponding operator, and let [0, ) X X 0, ), t, x, y) pt, x, y) be the integral kernel corresponding to e th ) t 0 L l 2 mx)), where p,, ) > 0 is implied by the connectedness of X, b). Let us make the essential point of Lemma 2.1 clear: QΦ,0 is closable although, in general, this form need not come from a well-defined symmetric operator Γ l 2 m X, F ) with domain Γ c X, F ). For a precise statement in this context, set { ΓX, F ; b) := f f ΓX, F ), } bx, y) fy) y < for all x X and define a covariant formal difference operator H Φ,0 by H Φ,0 : ΓX, F ; b) ΓX, F ) H Φ,0 fx) = 1 bx, y) fx) Φ y,x fy) ), x X, mx) with the obvious notation Remark 2.2. {y y b x} H scal θ,0, H := Hscal 0,0 in the scalar situation of Lemma 2.3. a) If it holds that bx, y) 2 7) < for all y X, mx) then one has 8) 9) H Φ,0 [Γ c X, F )] Γ l 2 m X, F ), Γ l 2 m X, F ) ΓX, F ; b) Q Φ,0 f 1, f 2 ) = H Φ,0 f 1, f 2 m for all f 1, f 2 Γ c X, F ). b) If X, b) is locally finite, in the sense that for all y X there are at most only finitely many x X with bx, y) > 0, then one has 7) and 10) c) Set ΓX, F ) = ΓX, F ; b). Cb, m) := sup deg m,b x) = sup 1 mx) ) bx, y).

7 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 7 If Cb, m) <, then one has 7), and H Φ,0 with domain of definition Γ c X, F ) is bounded with H m Φ,0 2Cb, m) and H Φ,0 = H Φ,0. If Cb, m) =, then one either has H Φ,0 [Γ c X, F )] Γ l 2 m X, F ), or H Φ,0 with domain of definition Γ c X, F ) is unbounded. Remark Lemma 2.3 a) is optimal in the following sense: If 7) is not satisfied, then one has H[C c X)] l 2 mx) by Theorem 3.3 in [9]. 2. Note that in the scalar situation one has H scal θ,0 : CX; b) CX) H θ,0 scal fx) = 1 mx) {y y b x} bx, y) fx) e iθy,x) fy) ), x X. 3. More specifically, upon taking X = Z l, { 1, if y x bx, y) := R l = 1, 0, else m 1, we get the magnetic lattice Laplacian 11) H scal θ,0 fx) = {y y x R l=1} fx) e iθy,x) fy) ), x Z l, scal which is bounded in the sense of Lemma 2.3 c), with H θ,0 = Hθ,0 scal. Let us point out two recent papers in this context that deal with the spectral theory of operators that generalize 11) for θ = 0 into two directions: Firstly, in [8] the authors consider δ-perturbations of Ψ H) with Ψ C 1 0, ) such that Ψ > 0, and in [13] the authors replace Z l by an arbitrary periodic lattice and perturb the resulting operator with periodic potentials. Proof of Lemma 2.3. a) By Cauchy-Schwarz we get that 7) implies the second inclusion in 8), and Green s formula cf. Lemma 3.1 in [19]) shows that the first inclusion in 8) implies 9). It remains to prove that 7) implies the first inclusion in 8). To this end, let f Γ c X, F ). By writing f as a finite sum f = z 1 {z}fz), we can assume f = 0

8 8 BATU GÜNEYSU away from some z X. Then we have H Φ,0 f 2 = 1 m mx) bx, y)fx) 2 bx, y)φ y,x fy) {y y b x} {y y b x} x ) 2 2 bx, y) fx) mx) x + ) 2 2 bx, y) fy) mx) y ) 2 2 fz) 2 z by, z) + 2 fz) 2 bx, z) 2 z <. mz) mx) b) This is obvious. c) Let f Γ c X, F ) be arbitrary, and assume first that Cb, m) <. Then we have bx, y) 2 Cb, m) bx, y) <, mx) in particular, HΦ,0 is a well-defined linear operator in Γ l 2 m X, F ) with domain of definition Γ c X, F ), and H Φ,0 f, f m = Q Φ,0 f) 2 bx, y) fx) 2 x 2Cb, m) f 2 m x, which entails that H Φ,0 is bounded with operator norm 2Cb, m) see also [3] for the particular case H Φ,0 = H). The assertion H Φ,0 = H Φ,0 follows from H Φ,0 f, f m = Q Φ,0 f). Assume now that Cb, m) =, and that H Φ,0 [Γ c X, F )] Γ l 2 m X, F ). Then Q is unbounded from above by Theorem 8.1 in [10]. But in view of H Φ,0 f, f m = Q Φ,0 f) Q f ), where we have used Green s formula and fx) Φy,x fy) 2 y fx) x fy) y 2, we find that H Φ,0 is unbounded as well. Let us now take perturbations by potentials into account. Given a potential V on F X, that is, V ΓX, EndF )) is pointwise selfadjoint, we can define a symmetric sesqui-linear form Q V in Γ l 2 m X, F )

9 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 9 by Q V f 1, f 2 ) = V x)f1 x), f 2 x) ) x mx) on its maximal domain of definition. Whenever V as above admits a decomposition V = V + V into potentials V ± 0 fiberwise as self-adjoint operators) such that Q V is Q Φ,0 -bounded with bound < 1, that is, if there are 0 < C 1 < 1, C 2 > 0 such that Q V f) C 1 Q Φ,0 f) + C 2 f 2 m for all f DQ Φ,0), then the sesqui-linear form Q Φ,V := Q Φ,0 + Q V is densely defined, symmetric, closed and semi-bounded from below). The self-adjoint semibounded operator corresponding to Q Φ,V will be denoted with H Φ,V. Remark 2.5. Note that in the scalar situation of Remark 2.2, any potential can be identified with a function v : X R, and as a particular case of the above construction we get the quadratic form Q scal θ,v in l 2 mx). The corresponding operator will be written as Hθ,v scal. One has: Proposition and definition 2.6. a) If a potential V on F X admits a decomposition V = V + V into potentials V ± 0 such that Q V is Q-bounded with bound < 1, then Q V is also Q Φ,0 -bounded with bound < 1, in particular, H Φ,V is well-defined. In this case, we say that V is Q-decomposable, and that V = V + V is a Q-decomposition of V. b) If a potential V on F X admits a decomposition V = V + V into potentials V ± 0 such that Q V is infinitesimally Q-bounded, that is, if for any ε > 0 there is a Cε) > 0 such that Q V f) εqf) + Cε) f 2 m for all f DQ), then Q V is also infinitesimally Q Φ,0 -bounded, in particular, H Φ,V is well-defined. In this case, we say that V is infinitesimally Q-decomposable, and that V = V + V is an infinitesimal Q-decomposition of V. Proof. Both statements follow from the fact that for any f DQ Φ,0 ) one has f DQ) with Q Φ,0 f) Q f ) cf. Theorem 2 ii) in [7]). Remark If a potential V on F X admits a decomposition V = V + V into potentials V ± 0 such that V KQ), the Kato class of Q for example if V C for some constant C > 0), then V = V + V is an infinitesimal Q-decomposition of V cf. Theorem

10 10 BATU GÜNEYSU 3.1 in [23]). Recall here that w : X C is in KQ), if and only if t lim sup ps, x, y) wy) my)ds = 0. t If V = V + V is an infinitesimal Q-decomposition of a potential V on F X, then V/ = V + / V / is an infinitesimal Q-decomposition of V/ for any > 0, in particular, H Φ,V/ is welldefined. We refer the reader to [19] for problems concerning the explicit domain of definition of H Φ,V, and essential-selfadjointness questions related with H Φ,V The semiclassical limit of the quantum partition function. We can now formulate the main results of this paper. Firstly, we consider the scalar nonmagnetic situation: Theorem 2.8. Assume that w : X R is infinitesimally Q-decomposable. Then for all β > 0 one has tr ) 12) e Hscal 0,w/ e βwx) [0, ] for all > 0, 13) lim inf 0+ in particular, 14) tr ) e Hscal 0,w/ e βwx), tr e 0,w/ ) Hscal 0+ e βwx). Note that we do not assume anything on X, b, m) here, and moreover, that we do not assume anything on v other than H0,w/ scal be welldefined: Indeed, as the formulation of Theorem 2.8 indicates, the limit of tr e 0,w/ ) Hscal as 0+ always exists, regardless of the fact whether or not one has e βw <. This existence heavily requires that no magnetic effects are present, which allows us to use arguments that rely on positivity preservation. We immediately get the following Weyl-type geometric result: Corollary 2.9. If the function m : X R, x lnmx)) is infinitesimally Q-decomposable for example, if m C for a constant C > 0), then one has tr e Hscal 0, m/ ) 0+ mx) [0, ].

11 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 11 We return to the general vector bundle setting. Here, the following can be said: Theorem Assume that the potential V on F X is infinitesimally Q-decomposable, and that there is an infinitesimally Q-decomposable function w : X R with V w. Then for all β > 0 with e βwx) < one has 15) tre H Φ,V / ) tr x e βv x) ) <, 16) tre βh Φ,V / ) 0+ tr x e βv x) ). Again, we do not assume anything on X, b, m) here, but now we cannot use monotonicity arguments to guarantee the inequality in 15), which requries the lower spectral bound V w for some suitable w with e βw <. Remark If V is an infinitesimally Q-decomposable potential on F X, then w := min specv )) : X R is easily seen to define an infinitesimally Q-decomposable function with V w, so that this part of the required control on V is not restrictive. In the case of scalar magnetic operators, Theorem 2.10 can be written in the following compact form: Corollary Let θ : {b > 0} [ π, π] be a magnetic potential, and let w : X R be infinitesimally Q-decomposable. Then for all β > 0 with e βwx) < one has tr e Hscal θ,w/ ) e βwx) <, tr e θ,w/ ) Hscal 0+ e βwx). 3. Proof of Theorem 2.8 and Theorem 2.10 We start by recalling some probabilistic facts from [6, 7]: Let Ω, F, P) be a fixed probability space, let Y n ) n N0 be a discrete time Markov

12 12 BATU GÜNEYSU chain with values in X such that bx, y) 17) PY n = x Y n 1 = y) = deg 1,b x) for all n N. Let ξ n ) n N0 be a sequence of independent exponentially distributed random variables with parameter 1, which are independent of Y n ) n N0. For n N set 1 S n := deg m,b Y n 1 ) ξ n, τ n := S 1 + S S n, where τ 0 := 0, so that we get the predictable stopping time τ := sup{τ n n N 0 } > 0, which allows us to define the maximally defined, right-continuous process X: [0, τ) Ω X, X [τn,τ n+1 ) Ω := Y n, n N 0, which has the τ n s as its jump times and the S n s as its holding times. Ultimately, with P x := P X 0 = x) and F = F t ) t 0 the natural filtration of F given by X, it turns out that Ω, F, X, P x ) ) is a reversible strong Markov process. Let us introduce the process Nt) := sup{n N 0 τ n t} : Ω N 0 { }, which at a fixed time t 0 counts the jumps of X until t, so that 18) Then we have: {Nt) < } = {t < τ} for all t 0. Lemma 3.1. The following statements hold for all t 0, x, y X, P ) 19) bx τn, X τn+1 ) > 0 for all n N 0 = 1, 20) pt, x, y)my) = P x X t = y), 21) P x Nt) = 0) = e deg m,b x)t. Proof. The relation 19), which simply means that X can only jump to neighbors, follows immediately from the definitions, 20) is well-known and follows from example from the Feynman-Kac formula for e th [6]), and 21) follows from P x Nt) = 0) = P x t < τ 1 ) = P x deg m,b x)t < ζ 1 ) = e deg m,b x)t, which follows immediately from the definitions. For x, y X and t > 0 let P x,y t be the probability measure on {t < τ} given by P x,y t = P x X t = y). With these preparations, we can state the following result, which the proof of Theorem 2.8 relies on:

13 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 13 Theorem 3.2. Assume that w : X R is Q-decomposable. Then the following assertions hold true: a) For any t > 0 one has 22) tr e thscal 0, w ) = b) For any t > 0 one has 23) tr e thscal 0, w pt, x, x)e x,x t [e ] t 0 wxs)ds mx) [0, ]. ) e t wx) [0, ]. Proof. This is precisely Theorem 5.3 in [6]. Here, it should be noted that the proof of the Golden-Thompson inequality 23) is highly nontrivial. In particular, 23) does not follow directly from 22). Now we can give the actual proof of Theorem 2.8: Proof of Theorem 2.8. Applying 23) with w = w/ keeping Remark in mind) and t = immediately implies tr ) e Hscal 0,w/ e βwx). In order to see that lim inf 0+ tr... ) is bounded from below by e βw, we remark that applying 22) with w = w/ and t = keeping using 20) in mind) implies the first equality in tr e Hscal 0,w/ ) = = = = = P x X = x)e x,x [ e 1 ] 0 wx s)ds [ P x X = x)e x,x 1 {N)=0} e 1 ] 0 wx s)ds P x X = x)p x,x N) = 0)e βwx) P x X = x) Px N) = 0, X = x) e βwx) P x X = x) P x N) = 0)e βwx) e deg m,b x) e βwx), where we have used 21) for the last equality. Thus, from Fatou s lemma we can conclude lim inf tr ) e Hscal 0,w/ e βwx), 0+

14 14 BATU GÜNEYSU which completes the proof. Before we come to the proof of Theorem 2.10, we first have to explain the additional ingredients of the Feynman-Kac formula for e th Φ,V : Firstly, using 18) and 19), the Φ-parallel transport along the paths of X is well-defined by // Φ : [0, τ) Ω F F = HomF y, F x ) 1 X0, if Nt) = 0 // Φ t := Φ XτNt) 1,X τnt) Φ Xτ0,X τ1 x,y) X X else HomF X0, F Xt ), which gives a pathwise unitary process, and furthermore we have the process A Φ,V : [0, τ) Ω EndF ) which is given by the path ordered exponential A Φ,V t = 1 X0 1) n n=1 EndF X0 ), where // Φ, 1 s 1 tσ n V X s1 )// Φ s 1 // s Φ, 1 n V X sn )// Φ s n ds 1 ds n tσ n := { s 1, s 2,..., s n ) 0 s 1 s 2 s n t } R n denotes the t-scaled standard n-simplex, and where the series converges pathwise absolutely and locally uniformly in t. With these preparations, we have the following covariant anaologue of Theorem 3.2: Theorem 3.3. Assume that the potential Ṽ on F X is Q-decomposable. Then the following assertions hold true: a) For any t > 0 one has tre th Φ,Ṽ ) = [ ] 24) pt, x, x)e x,x t A Φ,Ṽ t // Φ, 1 t mx) [0, ]. b) If there is a Q-decomposable function w : X R with e wx) <, Ṽ w and

15 SEMICLASSICAL LIMITS ON INFINITE GRAPHS 15 then for any t > 0 one has 25) tre th Φ,Ṽ ) tr x e tṽ x) ) <. Proof. Noting that pt, x, x)mx) 1 by 20), these results follow immediately from Theorem 4 in [7]. Now we are prepared for the proof of Theorem 2.10: Proof of Theorem Applying 25) with Ṽ = V/, w = w/ keeping again Remark in mind) and t = directly gives 15). In order to see 16), we apply 24) with Ṽ = V/ and t = to get the first equality in 26) tre H Φ,V / ) = = = = + [ p, x, x)e x,x [ P x X = x)e x,x tr x A Φ,V/ tr x A Φ,V/ [ E x Φ,V/ ) ] 1 {X =x}tr x A // Φ, 1 ) ] // Φ, 1 mx) ) ] // Φ, 1 [ E x Φ,V/ ) ] 1 {N)=0} tr x A // Φ, 1 [ E x Φ,V/ ) ] 1 {N) 1} 1 {X =x}tr x A // Φ, 1 In view of [ E x Φ,V/ ) ] 1 {N)=0} tr x A // Φ, 1 = P x N) = 0)tr x e βv x) ), the first summand in 26) tends to tr e βv ) ) as 0+, using 21) in combination with tr e βv ) ) < and dominated convergence. Finally, the second summand in 26) tends to 0 as 0+: To see this, recalling ν = rankf ) and using Proposition 6 i) in [7] a Gronwall type upper bound on the operator norm of A Φ,V/ ) we can estimate as

16 16 BATU GÜNEYSU follows, E [1 x Φ,V/ ) ] {N) 1} 1 {X =x} tr x A // Φ, 1 ν ν = ν E [1 x {N) 1} 1 {X =x} E x [1 {N) 1} 1 {X =x}e 1 E x [1 {X =x}e 1 ν E [1 x {N)=0} e 1 = ν tr ) e βhscal 0,w/ ν A Φ,V/ ] 0 wx s)ds ] x ] 0 wx s)ds ] 0 wx s)ds e deg m,b x) e βwx), which goes to zero by Theorem 2.8, as the second summand goes to ν e βw by dominated convergence. References [1] S. Albeverio, R. Høegh-Krohn, H. Holden and T. Kolsrud, A covariant Feynman Kac formula for unitary bundles over Euclidean space, in Stochastic Partial Differential Equations and Applications II Trento, 1988), Lecture Notes in Math., Vol Springer, Berlin, 1989), pp [2] Chung, F. R. K. & Sternberg, S.: Laplacian and vibrational spectra for homogeneous graphs. J. Graph Theory ), [3] Davies, E. B.: Large deviations for heat kernels on graphs. J. London Math. Soc. 2) ), no. 1, [4] Dodziuk, J. & Mathai, V.: Kato s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel. Contemp. Math. 398, Amer. Math. Soc. 2006), [5] Güneysu, B.: On generalized Schrödinger semigroups. J. Funct. Anal ), [6] Güneysu, B., Keller, M. & Schmidt, M.: A Feynman Kac Itô formula for magnetic Schrödinger operators on graphs. arxiv: v1. [7] Güneysu, B. & Milatovic, O. & Truc, F.: Generalized Schödinger semigroups on infinite graphs. Potential Anal 2013). DOI /s [8] Hiroshima, F. & Lörinczi, J.: The Spectrum of Non-Local Discrete Schroedinger Operators with a delta-potential. arxiv: [9] Keller, M. & Lenz, D.: Dirichlet forms and stochastic completneness of graphs and subgraphs. J. Reine Angew. Math ),

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