Correlation at Low Temperature: I. Exponential Decay

Transcription

1 Correlation at Low Temperature: I. Exponential Decay Volker Bach FB Mathematik; Johannes Gutenberg-Universität; D Mainz; Germany; Jacob Schach Møller Ý Département de Mathématique; Université Paris-Sud; F Orsay Cedex; France; April 22, 2002 Abstract The present paper generalizes the analysis [18, 2] of the correlations for a lattice system of real-valued spins at low temperature. The Gibbs measure is assumed to be generated by a fairly general pair potential (Hamiltonian function). The novelty, as compared to [18, 2], is that the single-site (self-) energies of the spins are not required to have only a single local minimum and no other extrema. Our derivation of exponential decay of correlations goes through the spectral analysis of a deformed Laplacian closely related to the Witten Laplacian studied in [18, 2]. We prove that this Laplacian has a spectral gap above zero and argue that this implies exponential decay of the correlations. MSC: 82B05, 81Q20. Keywords: Lattice Spin Systems, Witten Laplacian, Gibbs Measure, Exponential Decay of Correlations. Supported by TMR grant FMRX Ý Supported by Carlsbergfondet and partially supported by TMR grant FMRX

2 BM-April 22, Contents I Introduction 2 I.1 Organization of the Paper I.2 Assumptions and the main result I.3 Twisted Witten Complex I.4 Correlation Identity and Exponential Weights I.5 Comparison Operator I.6 Exponential Decay of Correlations II Form Bound on the Perturbation 23 II.1 Estimate on Ï Ó«Óµ in Terms of Âa µ II.2 Estimate on Ï Ó«Óµ Ï in Terms of Âa µ II.3 Estimate on Âa µ in Terms of Ãb II.4 Estimate on Ãb in Terms of Äb II.5 Estimate on Äb in Terms of Åb III Semiclassical Localization of the Partition Function 39 III.1 The Dilated Interaction III.2 The Dilated Hamiltonian Function A Construction of and from 45 B Some Basic Properties 51

3 BM-April 22, I Introduction The present paper is devoted to the study of probability measures on Ê given in form of a Boltzmann weight, ܵ ½ À ܵ Ü (I.1) where is a large, but finite set, À Ê Êµ is twice differentiable, and ½ is a normalization factor. In the context of (classical) statistical mechanics, measures of the form (I.1) are called lattice gases or lattice spin systems. In this part of theoretical physics, Ê Ü Ê is intepreted as the space of configurations of continuous spins on a lattice, and À Ê Êµ is a Hamiltonian function whose values À ܵ represent the energy of a given spin configuration Ü. The properties of (an appropriate sequence labeled by of) measures of the form (I.1) in the limit ½ is called the thermodynamic limit and is of central importance to statistical mechanics, as statistical mechanics deals with macroscopic properties of these physical systems for large lattices. In contrast, statements about (I.1), whose validity requires to be bounded above by a constant, are useless for most applications in statistical mechanics. (Exceptions are, e.g., spin glass systems.) For this reason, we insured that all statements in this paper are meant to hold uniformly in, unless explicitly stated otherwise. Moreover, corresponds to inverse temperature, Ì µ ½, and our interest is focused on the low-temperature regime, ½. (Here is Boltzmann s constant.) The measures (I.1) are studied in this paper by means of a twisted Witten Laplacian ÀÉ which is a selfadjoint operator acting on (a dense domain in) the space À Î Ê µ of square-integrable, antisymmetric forms on Ê. Denoting by the usual exterior differentiation on À, we introduce the twisted exterior complex as the conjugation of by certain exponential weights, ÀÉ ½ À ɵ À ɵ ½ ɵ À À ɵ (I.2) Here, É is a suitably chosen matrix-valued function on Ê, and the twist ɵ denotes its second quantization. Clearly, ÀÉ. Moreover, ÀÉ À because ɵ restricted to -forms vanishes. Denoting by ÀÉ the adjoint of ÀÉ with respect to Lebesgue measure, we introduce a twisted Dirac operator by ÀÉ ÀÉ ÀÉ. The twisted Witten Laplacian (or Hodge Laplacian with respect to the complex ÀÉ ) is then defined as the square of ÀÉ, ÀÉ ÀÉ ÀÉ ÀÉ ÀÉ ÀÉ (I.3)

4 BM-April 22, By construction, ÀÉ is a positive operator. Its kernel is one-dimensional and spanned by the -form À, as we show below. The (untwisted, É ) Witten Laplacian À was originally introduced by Witten [22] in the context of differential geometry. Its importance for the study of (I.1) lies in the identity Ì Ù Úµ Ù Úµ Ùµ Úµ (I.4) ½ Å À Ù ½µ À ½ À Ú «for the truncated Ê correlation function of the probability measure ܵ, where Ùµ ٠ܵ ܵ denotes the expectation value of a continuously differentiable observable Ù ½ Ê Êµ, and ½µ À denotes the (strictly positive) restriction of À to the space of ½-forms. Identity (I.4) shows that the decay (mixing property) of ܵ corresponds to the decay of the resolvent of À. This was realized by Helffer and Sjöstrand [15] and, more explicitly, by Sjöstrand in [18]. In particular, the truncated two-point correlation function has an explicit representation as a matrix element of the resolvent of À, Ì Ü Ü µ ½ Å À Ü ½µ À ½ À Ü «(I.5) Here, are two lattice points. The asymptotics of the decay of this truncated correlation function Ì Ü Ü µ, as the distance of and gets large, is of special interest. Eq. (I.5) links the asymptotics, in the low-temperature regime, to matrix elements of the inverse of À µ. To realize this connection, we need to analyze the low-lying spectrum of ½µ À. In this paper we prove an upper bound on the correlations and in a future work we will derive the asymptotics of the correlations, following ideas of [2]. A certain class of Hamiltonian functions were studied by means of ½µ in À the above mentioned paper by Sjöstrand [18]. He derived the asymptotics of the truncated correlation function in leading order in ½ under the assumption that the Hamiltonian function À is uniformly strictly convex. In [20], he considerably refined his analysis and derived the correlation asymptotics to all orders in ½. Sjöstrand s paper [18] was also the starting point for Jecko, Sjöstrand, and the first author, who derived in [2] the (same) correlation asymptotics, to leading order in ½, under weaker assumptions. Yet, the the requirements in [2] still imply that À has no critical point except for its (unique) minimum. While heuristic arguments suggest that uniqueness of the global minimum of À is actually a necessary condition for exponential decay of the correlations, the exclusion of other critical points in [2] appeared to be purely technical and not essential. This is precisely the point to be established in the present paper. Loosely speaking, the

5 BM-April 22, introduction of the twist ɵ convexifies the potential in ½µ ÀÉ such that it has all properties required in [2], and all the methods developed therein apply again. Apart from this (desired) convexification, ɵ also generates (undesired) exponentially large weight factors. These large weights lead to technical complications in our analysis because we have to show that they do not affect the main relative form bound implying the expected spectral properties of ½µ ÀÉ (see Sect. II). The twist with ɵ leads us to analyze the stability at low temperatures of the partition function, under perturbations of À in a single variable, more precisely of the type À À Õ, where Õ Üµ Õ Ü µ and Õ vanishes around. We prove that pertubations of this type only change by an error which is exponentially small in. We are inspired by techniques of Sjöstrand [19], cf. also [10, 16]. A variety of Hamiltonian functions, especially those of -type, were analyzed. He also uses a twist, which is, however, different from ours. Nevertheless, Helffer s approach and the models he studied do overlap with the models treated in the present paper. Therefore, a comparison is appropriate and will be the subject of future research. by Helffer [11, 13, 12, 14] by means of the Witten Laplacian ½µ À It turns out that, through (I.3), the spectral properties of ½µ À are also intimately connected with the validity of a Log-Sobolev inequality for the measure. Roughly speaking, a Log-Sobolev inequality implies the uniform (w.r.t. ) positivity of ½µ À, and the converse is true, as well, provided the positivity of ½µ À is not only uniform w.r.t., but also with respect to all possible boundary conditions (in the sense of the DLR equations). This has been established by Zegarlinski [25], Yoshida [23], Bodineau and Helffer [3] and Procacci and Scoppola [17]. These works, in particular, prove exponential decay of correlations for weakly coupled systems at an arbitrary, but fixed inverse temperature ½. In contrast to our paper, however, the methods employed in these papers do not apply in the low-temperature limit ½. The paper [3] is close to ours in spirit whereas [17] uses cluster expansion techniques. One of the interesting aspects about the Witten Laplacian is that it appears in various branches of mathematics and mathematical physics as an object of fundamental importance. Aside from the original application to problems in differential geometry, now sometimes referred to as Witten-Helffer-Sjöstrand theory [5], it coincides (on a formal level) with operators used in supersymmetry, cf. [7]. In stochastic analysis, the Witten Laplacian can be viewed as the generator of a parabolic, stochastic PDE that determine the correlation functions of and the rate of convergence of initial distributions to a stationary one [1]. In the context of spin glasses, especially the Hopfield-Kac model, the Witten Laplacian has been used in [4], to estimate the tunneling rates between local minima of a spin glass Hamiltonian.

6 BM-April 22, We remark that continuous spin systems with multiple phases have been successfully studied by other methods, e.g., the Pirogov-Sinai theory and contour methods [8, 9, 24]. It would be interesting to link the basic notions of Pirogov- Sinai theory to corresponding spectral properties of the Hamiltonian. We anticipate this link to go through the minima of the Hamiltonian function which, on one hand, counts the number of phases in Pirogov-Sinai theory, and which, on the other hand should correspond to the elements of the kernel of a Witten Laplacian which degenerates in the thermodynamic limit. This is a subject of future research, as well. Acknowledgements: We would like to thank B. Helffer, T. Jecko, O. Matte and J. Sjöstrand for useful discussions and the second author would like to thank E. Skibsted and Aarhus University for hospitality.

7 BM-April 22, I.1 Organization of the Paper The present section is devoted to a presentation of all the objects used in this paper. We furthermore state the main result, Theorem I.1, and reduce its proof to two statememts. One is a semi-classical localization result, Theorem I.6, and the other is a form bound given in Theorem II.1. In Subsect. I.2 we formulate the assumptions we impose on the self-energies (or single-spin potentials) and the interaction between lattice sites. We furthermore present some properties of the self-energies and the interaction which follow from the basic assumptions (to be derived in Appendix A). In Subsect. I.3 we present the central object around which the analysis of this paper revolves, namely the twisted Witten Laplacian (I.3). In Subsect. I.4 we present an extension to non-zero É of the identity (I.4) expressing the truncated two-point correlation function in terms of matrix elements of the inverse Witten Laplacian (restricted to ½-forms). We furthermore give a Combes- Thomas twisted version of this formula which will be used to prove exponential decay of correlations. In Subsect. I.5 we introduce a comparison operator which approximate the twisted Witten Laplacian in the low-temperature limit (in a semiclassical sense). In Subsect. I.6 we state and prove the main result of this paper, namely exponential decay of correlations. In Sect. II we state and prove Theorem II.1, the main technical result of this paper. This theorem is a form bound which states that the twisted Witten Laplacian (with Combes-Thomas weights), in the low-temperature limit, is a small pertubation of the comparison operator to be introduced in Subsect. I.5. We start the section by breaking the proof up into a series of estimates, which are subsequently proven in a series of subsections. In Subsect. II.1 and II.2 we estimate the perturbation (which is a matrix) in terms of a diagonal matrix. In Subsects. II.3 II.5 this diagonal matrix is in turn estimated by other diagonal matrices which in the final step is identified with the diagonal part of the comparison operator. In Sect. III we prove a semi-classical localization estimate for the partition function. The proof is divided into two steps. In Subsect. III.1 we analyze the interaction part of the Hamiltonian and in Subsect. III.2 we analyse the self-energy part of the Hamiltonian and prove the localization estimate. In Appendix A we derive some properties of the self-energy and the interaction which follow directly from the basic assumptions. Finally, in Appendix B we establishe self-adjointness of the Dirac operator and determine the kernel of the Witten Laplacian. I.2 Assumptions and the main result We consider a system of real-valued spins on a finite set, i.e., ½, which we henceforth refer to as the lattice. We assume to be a metric space with

8 BM-April 22, respect to a metric Ê. Typical examples for we have in mind are finite subsets of or finite discrete tori Ä Äµ, with Ä Æ. While Ä is an abelian group, we shall not assume to possess any additional structure at this point. In order to allow for taking the thermodynamic limit, we shall think of the lattice as being an element of a countable collection of finite lattices,. For the existence of the thermodynamic limit and its analysis, we then require that all constants are uniform with respect to. Below, we call a quantity universal if it neither depends on nor on. For example, we shall assume below that ÑÜ µ (I.6) for some universal constant ½. Given the lattice, the corresponding spin configuration space is Ê, and the energy of a spin configuration is determined by a pair interaction Hamiltonian function À Ê Ê given by À ܵ Ü µ «Û Ü Ü µ (I.7) for Ü Ü µ Ê. Here, Ê Ê are the on-site energy functions at the lattice point, and Û Ê Ê represent the pair interaction between the lattice sites. Furthermore, ««½ is a coupling constant which is assumed to be sufficiently small, although we do not consider the limit «. For the derivation of our main results, we require the following hypotheses. Hypothesis 1. For any, zero is the unique, nondegenerate minimum of Ê Êµ, attained at Ø, i.e., µ, µ, and ص, whenever Ø. Moreover, there exist universal constants ½ ½, Ê ½, such that for all. Ø Ø Ê µ ص µ ص (I.8) (I.9) ص ص ÑÒ½Ø (I.10) (I.11) Ø Ê Øµ ÑÜ µ Ø Hypothesis 1 roughly coincides with what is required in [2, Example I.1], except for Eq. (I.10), which is much weaker than the corresponding assumption ص µ ÓÒ Ø Øµ ص (I.12)

9 BM-April 22, in [2, Example I.1]. Note that Eq. (I.12) yields Ø µ µ, for any critical point Ø, which implies that Ø is the only critical point of. This does not only suggest that Eq. (I.12) is merely technical, but it also illustrates that the requirement (I.12) excludes one of the main examples for an application of our method: an Ising-type or -model in a magnetic field, in which case Ü µ ÓÒ Ø Ü ½µ Ü, with Ê being the magnetic field at site. In fact, the main goal of the present work is to weaken Assumption (I.12). To formulate our hypothesis on Û, we introduce functions for all. µ ÑÒ µ µ ½ ÑÒ½ (I.13) Hypothesis 2. For all, the pair interaction functions Ê Ê Êµ vanish on-site and at the origin, Û Øµ and Û µ. Furthermore, for, there exists a symmetric matrix µ of nonnegative weights (with for convenience), such that Û Ü Ü Ü µ Û Ü Ü Ü Ü µ Û Ü Ü µ Ü µ Ü µµ (I.14) and Û Ü Ü µ (I.15) The weights are assumed to have exponential decay with respect to the metric on, µ (I.16) ÑÜ for some universal constant ½. Similar to the example above, Hypothesis 2 also accommodates for the important example of an Ising or spin glass type interaction, for which Û Ü Ü µ Ü Ü (except that may not be universal for spin glass models). Further note that in the case and the euclidean metric, the summability condition is easily fulfilled for interactions of finite range, i.e., in case that Û whenever µ. Hypotheses 1 and 2 insure that À is growing at least linearly in Ü at infinity, provided «is sufficiently small, cf. Lemma B.1. Thus Ù ÜÔ À Ä Ê µ, for any and any polynomially bounded measurable function Ù

10 BM-April 22, Ê. In particular, the expectation value Ùµ in Eq. (I.1) exists for any such function (which represents an observable). The main result of this paper is the following. Theorem I.1 (Exponential Decay). Assume Hypothesises 1 and 2. There exist universal constants ½ and «½ such that, for all ½, ½, ½, and ««½ µ«½, we have Ì Ü Ü µ ½ µ ½ ÑÒ µ (I.17) Here ÑÒ ÑÒ À µµ which, for ««small enough, satisfies ÑÒ. We remark that the effect of boundary conditions (in a given example) on the decay estimate (I.17) is encoded in ÑÒ and in, and hence, is easy to analyze. Control with respect to boundary conditions were considered in [25, 23, 3, 17], however, not with uniform control at low temperature. An important point in our analysis is the observation made in Lemma I.2, below, that our hypothesis on suffices to construct an auxilliary function which obeys (I.12) and, yet, differs from only away from its global minimum at Ü, i.e., in a region which should be irrelevant (and, indeed, is irrelevant, as we prove in a subsequent paper) for the correlation asymptotics at low temperature. Lemma I.2. Assume Hypothesis 1. There exist universal numbers Õ ÑÜ, Ê Ê ½ Ê Ê ½ ½ (I.18) to which we associate the unions of intervals and functions Á Ê Ê (I.19) Á Ê ½ Ê µ Ê Ê ½ µ (I.20) Á ½Ê µ Ê ½µ (I.21) Á ½ ½Ê ½ µ Ê ½ ½µ (I.22) Õ Á Õ ÑÜ Õ Á Õ ÑÜ (I.23) and Õ Õ Õ Ê Õ ÑÜ (I.24) possessing the following properties: (i) On Á ½, we have Õ Õ ÑÜ. (ii) The functions Õ and Õ (I.25) are nonnegative and have a unique critical point at Ø.

11 BM-April 22, (iii) There exist universal constants ½ ½, such that the following estimates hold true: µ µ (I.26) Ø Ê Øµ ÑÒ½Ø (I.27) Ø Ê Ø Ê Øµ µ ص ص ÑÜ Ò µ Ø Ó µ ص (I.28) (I.29) Ø Ê Øµ ص ص (I.30) ÑÜ Ø Øµ (I.31) ÑÒ Ø Øµ ÑÒ½ (I.32) Although in Appendix A, we give a detailed proof of Lemma I.2, the reader might find our graphical illustration in Fig. 1 of its assertion for a typical function helpful. Note that an application of Lemma I.2 to Hypothesis 2 easily yields the following consequence. Lemma I.3. Assume Hypotheses 1 and 2. Then, there exists a universal constant Û ½, such that, for all, the pair interaction functions Û obey Û Ü Ü Ü µ Û Ü Ü Ü Ü µ Û Ü Ü µ (I.33) Û ÑÒ Ü µ Ü µ ½ ÑÒ Ü µ Ü µ ½ I.3 Twisted Witten Complex In this section we present most of the objects to which we reduce the analysis of the truncated two-point correlation functions (I.2). We introduce the fermionic Fock space over, µ ½Ò Òµ Òµ Å Ò (I.34) where Å Ò denotes the Ò-fold antisymmetric tensor product, and µ ³ Å is a one-dimensional subspace spanned by the normalized vacuum vector Å. The standard annihilation and creation operators represent the canonical anticommutation relations (CAR) Æ and Å (I.35)

12 BM-April 22, Ü µ Õ ÑÜ ÑÜ Õ Õ ÑÜ Õ Õ Ü Ê Ê ½ Ê Ê ½ Figure 1: The solid curve is the graph of. The graph of is the dashed curve on Ê Ê ½ µ and agrees with on Ê µ and on Ê ½ ½µ. The graph of is the dashed curve on Ê Ê ½ µ, the dash-dotted curve on Ê ½µ and agrees with on Öµ and on Ê ½ Ê µ. The dash-dot-dotted curve depicts Õ, and the dash-dotdot-dotted curve depicts Õ

13 BM-April 22, on, where, and Æ is the Kronecker delta. By (I.35), an orthonormal basis of ½µ is given by Å. Thus, µ and ½µ can be identified with and, respectively. Due to (I.35), the products and leave Òµ invariant. Thus, we can represent their action on µ by a complex number, and Æ on µ (I.36) Likewise their action on ½µ can be represented by a complex -matrix. Denoting the standard matrix units on ½µ by, i.e., Å Ð Å Æ Æ Ð, one easily checks that and Æ ½ on ½µ (I.37) Note, in particular, that and µ on ½µ (I.38) È È Here, denotes µ Ò. This is part of the summation convention that È È È È µ µ, µ È µ µ, µ È µ È È Ò µ µ, µ µ Ò µ, which turns out to be convenient and is used and throughout the paper. The Hilbert space of forms over Ê is the tensor product À Ä Ê µ Å ½ÒÀ Òµ À Òµ Ä Ê µ Å Òµ (I.39) We introduce a (multiple of the) standard exterior derivative on À, for, and its adjoint ½ Å and ½ Å (I.40) where is shorthand for Ü. Note that µ. The Hodge Laplacian associated to this exterior derivative is µ. To a bounded operator Ì À ½µ we associate its second quantization Ì µ À by the standard formula, i.e., if Ì is represented by a matrix Ì µ whose entries take their values in the operators on Ä Ê µ then Ì µ Ì Å (I.41)

14 BM-April 22, In the present paper we second-quantize only operators whose entries are semibounded self-adjoint operators on Ä Ê µ. In particular, let É, É, and É denote the matrix-valued functions with entries É Üµ Æ Õ Ü µ É Üµ Æ Õ Ü µ (I.42) and É Üµ É Üµ É Üµ (I.43) Here, Ü Ü µ Ê, and Õ and Õ are introduced in (I.23) (I.24). We frequently omit the argument and write Õ Õ Ü µ and Õ Õ Ü µ, et cetera. Then their second quantization is given by É µ Õ Å (I.44) where É denotes É, É, or É, and Õ denotes Õ, Õ, or Õ, respectively. Now, we introduce the twisted exterior derivative ÀÉ À ɵµ À ɵµ (I.45) on ½ Ê µ Å. Here, À À Å ½ is considered a multiplication operator on À. The twisted Dirac operator is defined as the sum of the twisted exterior derivative and its adjoint, ÀÉ ÀÉ ÀÉ (I.46) By construction it is clear that ÀÉ ÀÉ µ. Thus the square of the twisted Dirac operator is the associated Hodge Laplacian ÀÉ ÀÉ ÀÉ ÀÉ ÀÉ ÀÉ (I.47) which we will call the twisted Witten Laplacian. It was introduced by Witten (in the case É, À a Morse function, and Ê replaced by a compact manifold) in [22] to give a semiclassical proof of the Morse inequalities (see also [7]). The twisted Witten Laplacian preserves the space À Òµ of Ò-forms on Ê, so, writing Òµ ÀÉ for the restriction of ÀÉ to À Òµ, we have that Å ÀÉ Òµ ÀÉ Ò (I.48) In Appendix B, we discuss domain properties of ÀÉ, ÀÉ, ÀÉ, and ÀÉ, as well as, the question of self-adjointness of ÀÉ and ÀÉ. One of the results proved in Appendix B is the following. Theorem I.4. Assume Hypothesis 1 and Hypothesis 2.

15 BM-April 22, (i) There exist «and ½ such that, for ««and, ÃÖ µ ÀÉ À and ÃÖ ½µ ÀÉ (I.49) (ii) If all the s and s are smooth in (not necessarily universal) neighbourhoods of, then there exists a universal constant «such that for ««and all, ÃÖ ÀÉ À Å Å (I.50) We remark that our main result Theorem I.1 only requires (I.49). We have however included a proof in Appendix B of the more general (I.50). In the remaining part of this section we give more explicit formulas for the objects introduced above, and we compare these with [2]. First, we introduce some exponential weights, Õ Õ and Õ (I.51) and twisted derivatives, together with their adjoints, Àµ ½ À and Àµ ½ À (I.52) We compute, using (I.35) and (I.40) (I.45), Àµ Å and ÀÉ Àµ Å ÀÉ (I.53) Note the intertwining relations Àµ µ and Àµ µ (I.54) where À À Õ Õ ««Û Û (I.55) (I.56) according to (I.7). In the derivation of (I.54) we use that À Õ µ and À Õ µ. The intertwining relations (I.54) are of key importance to our analysis, as it allows us to pass from À to a new Hamiltonian function which agrees with À at zero, but yet has no critical points other than zero.

16 BM-April 22, Note that À µ with Ä Ê µ Å. We will use the same notation, µ ÀÉ Ä Ê µ Å µ can be identified with Ä Ê µ, and À ½µ and ½µ ÀÉ, for both realizations. We first compute the restriction of the twisted Witten Laplacian to -forms, µ ÀÉ, using (I.36) which implies that ÀÉ on À µ. We obtain µ ÀÉ ÀÉ ÀÉ Àµ Àµ Å µ µ Å ½ µ µ (I.57) where in the last line, we have given its representation on Ä Ê µ. In the special case É, we have ½, for all, and hence µ À Àµ Àµ ÖÀ ܵ ½ À ܵ (I.58) È where. For É, the twisted Witten Laplacian restricted to - forms thus becomes a standard semiclassical Schrödinger operator, and the critical points of À become global minima of the potential (if we disregard À, which enters at lower order in ). Comparing to (I.57), we see that the introduction of É removes the nonzero local minima, which are irrelevant in the low-temperature limit. The price we pay is the presence of the exponential weights. In [2] the assumptions were such that the low-temperature study of the Witten Laplacian, with É, became essentially a one-well semiclassical problem. Here we have relaxed the assumptions and introduced the extra twist by ɵ to restore the essential features of a one-well problem. The full twisted Witten Laplacian is of the form, cf. (I.35), (I.47), (I.53) and (I.54), ÀÉ Ò Àµ Àµ Å Àµ Àµ Å Ò Ó Àµ Àµ Å µ µ Å Àµ Àµ Å Àµ Àµ Å µ µ Å À ܵ Å Ó (I.59)

17 BM-April 22, where we use Àµ and Àµ Àµ ½ À, for, and we denote À À. Inserting (I.38) into (I.59), we arrive at the twisted Witten Laplacian on ½- forms, Ò Ó ½µ Àµ Àµ µ µ Å ÀÉ À Å µ I.4 Correlation Identity and Exponential Weights (I.60) A remarkable consequence of Theorem I.4 is that it allows us to write the truncated correlation (I.2) of two observables Ù Ú ½ Ê Êµ as a matrix element of the resolvent of ½µ ÀÉ, as follows. Theorem I.5. Assume (I.49) and let Ù Ú ½ Ê Êµ such that ÖÙ and ÖÚ are polynomially bounded in Ü. Then Ì Ù Úµ ½ (I.61) where Ùµ À Õ µ Å Å ½µ ÀÉ Proof: We first observe that Ùµ ½ and hence Ê À ܵ Ü Ù Üµ À ܵ Ü ½ Ê Ì Ù Úµ Ù Úµ Ùµ Úµ ½ Úµ À Õµ Å Å À Å Å (I.62) Å À Å Å «Ù À Å Å (I.63) Ù Ùµµ Ú Úµµ (I.64) Å Ù Ùµµ À Å Å «È Ú Úµµ À Å Å ½ where we use that Ú is real-valued and È ½ ½ À À Å È Å (I.65)

18 BM-April 22, denotes the orthogonal projection onto the orthogonal complement to À times the projection onto the Fock vacuum, È Å ÅÅ. We used above that Ù Ùµµ ÜÔ Àµ Å Å ÊÒ È. Next, Eq. (I.49) implies that ÊÒ ÀÉ È ÃÖ ÀÉ, which allows us to write È È ÀÉ ½ ÀÉ ÀÉ È È ÀÉ ½ ÀÉ ÀÉ È È ½µ ÀÉ ÀÉ ½ ÀÉ È (I.66) Using furthermore that ÀÉ Ùµ ÜÔ ÀµÅÅ, we obtain from Eqs. (I.64) and (I.66) that Ì Ù Úµ ½ A simple computation gives ÀÉ Ù À Å Å ½µ ½ ÀÉ ÀÉ Ú À Å Å (I.67) ÀÉ Ú À Å Å ½ Úµ À Õ µ Å Å (I.68) Inserting (I.68) into (I.64), we arrive at the desired identity (I.61) for the truncated correlation. As an immediate consequence of (I.61), we have the identity Ì Ü Ü µ ½ À Õ µ Å ½µ ÀÉ ½ À Õ µ Å (I.69) for the two-point correlation function, where Å denotes the Ø standard basis vector in. The presence of the modifiers Õ and Õ in the exponents leads us to study the stability of the partition function under perturbations in a single spin variable. We have the following theorem, which is proved in Section III Theorem I.6. Assume Hypotheses 1 and 2. There exists a universal constant such that, for all Ó,, ½, and ««ÑÒ½, the following localization estimates hold true, ½Ü Ó ½ ÜÜ Ó À ܵ Ü ½ ÑÒ (I.70) This result states that perturbing the Hamiltonian function in one spin-variable only changes the partition function by an exponentially small error in the lowtemperature regime, provided the perturbation is supported away from the global minimum. Exponential decay estimates for the measure ÜÔ À Ü, as a single spin variable is away from the global minimum are complemented by the decay properties of Ì Ü Ü µ in (I.69), as the distance between and in tends to be large. For the analysis of the latter, we introduce exponential weights ÜÔ µ and observe the following generalization of (I.69).

19 BM-April 22, Lemma I.7. Let and. Then µ Ì Ü Ü µ ½ À Õµ Å Å ½µ where ½µ ÀÉ µ denotes the restriction to À ½µ of ÀÉ µ ÀÉ µ ½ À Õ µ Å Å Ò Àµ Àµ Å µ µ Å µ µ À Å (I.71) Ó (I.72) and Ê is the metric on. Before we turn to the proof of Lemma I.7, we remark that the exponential decay of the two-point correlation function is analyzed in detail in Subsect. I.6. Loosely speaking, the decay rate of the two-point correlation function is given by the limes superior of all, for which (I.61) stays bounded, as µ gets large. Another remark to be made is that ÃÖ µ ÀÉ µ À and ÃÖ ½µ, for all ½ and, due to the fact that ÀÉ µ derives from ÀÉ ÀÉ µ by conjugation with the bounded and invertible operator ÜÔ Å µ, see below. Proof: Define a diagonal -matrix Å by Å µ µ Æ. Observe that Å µ Å µ Å and that Å µ leaves À Òµ invariant, cf. (I.41). Hence ÀÉ µ µ Ì Ü Ü µ (I.73) Since Å µ À Õµ Å Å ½ ÀÉ Å µ À Õµ Å Å À Õ µ Å Å ÅÓµ ÀÉ ÅÓµ ½ À Õ µ Å Å Å µ Šе we obtain Eq. (I.71) from the computation µ µ Å µ ÀÉ Å µ Ò Àµ Àµ Å µ µ Å µ µ À ܵ Å (I.74) Ó (I.75)

20 BM-April 22, I.5 Comparison Operator To motivate our next definitions, we observe that, by setting É in (I.60), we obtain the Witten Laplacian that has been analyzed in [2], ½µ À µ À Å ½ À ܵ (I.76) In [2], the semiclassical confining properties of the one-well problem were used to show that ½µ À is close to the comparison operator ½µ À µ Å ½ À À µ (I.77) in the sense that their difference Ï ½µ À ½µ ½µ À À is a small perturbation of, as a quadratic form. More specifically, it was shown in [2] that if «½µ À is sufficiently small and ½ is sufficiently large, then for any vector in the form domain of ½µ, À Å ½µ Ï «À ½ Å ½µ À «(I.78) for some universal constant ½. The derivation of Form Bound (I.78) in [2], however, uses Property (I.12), which we are lacking in the present context, in an essential way. To treat Hamiltonian functions that merely obey the weaker Hypotheses 1 and 2, we suppose we have constructed Õ and Õ according to Lemma I.2 and formed ÜÔÕ and ÜÔÕ, respectively, for all. We then define ÀÉ where Ò Å Å Ó À µ Å Àµ Àµ ½ ܵ (I.79) (I.80) µ µ (I.81) We remark that Eq. (I.79) coincides with (I.77), for É. (The notation is motivated by the estimate (I.92).) Furthermore, for ½ and Ó, we

21 BM-April 22, define Ï ÀÉ Ó Üµ Ï Üµ Ï Ó«Ó Üµ (I.82) Ï Üµ Ï Ó«Ó Üµ «Ï ܵ Ï Ó«Ó Üµ Ï Ó«Ó µ ܵ Å À µ Óµ Óµ À ܵ Óµ Óµ Û Üµ À µ Û µ Å (I.83) (I.84) Å Ï Ó«Ó Üµ Ï Ó«Ó Üµ Ï Ó«Ó µ (I.85) «Óµ Óµ Û Üµ Û µ Å Ï Ó«Ó µ «Óµ Óµ ½ Û µ Å where Û Û, and we observe the decomposition identity (I.86) ÀÉ Óµ ÀÉ Ï ÀÉ Óµ (I.87) The argument Ü in (I.82) (I.84) indicates that these operators act as matrixvalued multiplication operators and contain no differential operator. We frequently omit to display Ü. Note also that we have the identites Ï Ó«Ó µ and Ï Ó«Ó Üµ Ï Ó«Ó Üµ. For Ò ½, we compute the restrictions of ÀÉ onto À Òµ, which we denote by Òµ ÀÉ µ ÀÉ ½µ ÀÉ, respectively. From (I.36) (I.38), we immediately obtain µ µ (I.88) Àµ Àµ ½ ܵ µ µ µ We further observe that ÀÉ ½µ ÀÉ ÀÉ Å Å Ò µ (I.89) À µ Å Å Æ ½ (I.90) Ó

22 BM-April 22, where Æ ½µ (I.91) is the fermion number (= form degree) operator. Note that Æ is bounded, more precisely Æ, thanks to the Pauli principle. We will write Ï ÀÉ µ Óµ for the restriction of Ï ÀÉ Óµ to -forms, and likewise for the derived quantities presented in the formula (I.82). I.6 Exponential Decay of Correlations In this subsection we prove our main result, Theorem I.1. Our proof relies on a form bound to be presented below, cf. (I.94). This form bound is contained in Theorem II.1, which is the technical centerpiece of the present work. Proof of Theorem I.1: We begin with the remark that from (I.80), (I.54), and the facts that and ½, we get µ µ ½ µ µ ½ µ µ (I.92) In particular, this implies that, and hence we find that ½µ ÀÉ À µ ÑÒ ½ (I.93) where ÑÒ ÑÒ À µµ is the lowest eigenvalue of the Hessian of À at Ü. Observe that, since µ and Û µ, by Hypotheses 1 and 2, the eigenvalue ÑÒ is strictly positive (for ««small). Let (to be chosen below). Theorem II.1 implies that there exists a universal constant ½, such that, for all ½, ½, «½ ½ µ and all, the following estimate holds true Å Ï ½µ ÀÉ µ«½µ ÀÉ µ½ ½µ ÀÉ µ½ (I.94) Write ÀÉ µ ½ ½ ½ ½µ ½µ ½µ ÀÉ ÀÉ ½ Ï ½µ ÀÉ µ ½µ (I.95) ½ ½ ÀÉ ½µ ÀÉ ½

23 BM-April 22, The formbound (I.94), with ½, allows us to expand the resolvent on the righthand side of (I.95) in a Neumann series. This observation together with Eqs. (I.93) imply that for small enough (depending on ) we have ½µ ÀÉ µ ½ ½ µ ÑÒ (I.96) under the conditions on «and mentioned above and for all. Lemma I.7 and the Cauchy-Schwarz inequality yield Ì Ü Ü µ µ À Õ µ À Õ µ ½µ ÀÉ µ ½ (I.97) Note that the Hypothesis 1 and 2 remain true if we replace one of the s with the corresponding. We proceed to apply Theorem I.6, Ê À Õ µ Ü This implies, for all, À Ü À Õµ Ü Ü Ê Ü Ê Ê À Ü Ê À Õ µ Ü ½ µ ½ Ê À Õ µ Ü (I.98) (I.99) provided ½ is sufficiently large. Inserting (I.96) and (I.99) into (I.97), we arrive at the desired bound, Ì Ü Ü µ ½ ÑÒ µ We note that the formbound (I.94) implies the lower bound (I.100) ÑÒ ½µ ÀÉ ½ µ ÑÒ (I.101)

24 BM-April 22, II Form Bound on the Perturbation This section is the technical heart of our paper. Here we prove a form bound showing that ½µ ÀÉ ½µ ÀÉ Ï ½µ ÀÉ decomposes the twisted Witten Laplacian on À ½µ into its main part ½µ ½µ ÀÉ, see (I.89), and a small perturbation Ï ÀÉ. Being slightly more general, we show that Ï ÀÉ is a small compared to ½µ ÀÉ µ, displayed in (I.90), with relative bound of order ½, which becomes small in the low-temperature limit. The main result of this section is Theorem II.1 (Main Form Bound). There exist universal constants ½, «, and ½ such that, for all ««, all, all ½, and all Ó, the estimate ½µ ÀÉ µ ½ Ï ÀÉ Óµ ½µ ÀÉ µ ½ ½ ½ «½ (II.1) holds true on À µ, and ÏÀÉ Óµ on À µ. Especially, for and on À ½µ, we have Ï ÀÉ ½µ Óµ ½µ ½ ÀÉ (II.2) in the sense of quadratic forms. Proof: We break up the proof of Theorem II.1 into a series of lemmata proved in Subsects. II.1 II.5, below. Each of these lemmata contain a bootstrap argument close in spirit to the strategy applied in [2, Sect. III]. We recall from Eq. (I.90) the definition Æ È ½ Å Ä Ò Ò ½ À Òµ of the fermion number operator on À. We may assume without loss of generality that all operators are restricted to À µ. Recall the decomposition, see (I.82) (I.86), Ï ÀÉ Ó Üµ Ï Üµ Ï Ó«Ó Üµ Ï Ó«Ó µ (II.3) We estimate Ï Üµ Ï Ó«Ó Üµ and Ï Ó«Ó µ separately. In Lemma II.3 in Subsect. II.1 we prove that Æ ½ Ï Ó«Ó µ Æ ½ «½ µ (II.4) and the main part of the proof is actually to show that Å ½ Æ ½ Ï Ï Ó«Óµ Å ½ Æ ½ ½ (II.5)

25 BM-April 22, for some universal constant ½. Here, Å belongs to the following chain of (possibly unbounded) operators, Â Ä Â Å Ä Å Ã Å Ã Å Å Å (II.6) (II.7) where Â, Ã, Ä and Å are multiplication operators, multiplying by the following functions and  µ à µ Ä Å Î µ µ ΠΠܵ ܵ ½ ܵ and Πܵ ܵ ½ Furthermore, we denote the weights by µ, µ (II.8) (II.9) (II.10) (II.11) ܵ (II.12) µ µ, and introduce new weights µ, with the property that µ, in Subsect. II.3. To explain the general strategy of the estimates we note that controls the derivatives of Û, by Hypothesis 2. This enables us to show in Theorem II.6 in Subsect. II.3 that for some universal constant ½,  µ ½ Ï Ï Ó«Óµ  µ ½ (II.13) Next, we remark that the difference between and also contains only derivatives of Û which we control with. Bootstrapping the corresponding estimate, we derive in Theorem II.8 in Subsect. II.3 that, for some universal constant ½,  µ  à (II.14) For the bootstrap argument to work out, we need to pass from the weights µ µ to bigger weights µ. These new weights are then chosen

26 BM-April 22, as to fulfill È, where ½ is a universal constant. This construction and the bootstrap argument require «to be small. Similar arguments are then used in Theorem II.9 in Subsect. II.4 to prove that, for some universal constant ½, Ã ½ Ä Æ (II.15) provided «and are sufficiently small, and finally in Theorem II.10 in Subsect. II.5 to prove that, for some universal constant ½, Ä ½ Å Æ (II.16) provided that ½ is sufficiently large. Inserting (II.14) (II.16) into (II.13), we conclude that, for some universal constants ½ and all in the form domain of Å, Å Ï «Ó«Óµ Ï (II.17) Ò ½ Å Æ Ó Æ ½ Ò ½ Å Æ Ó Æ ½ Å ½ ½ Æ ½ Å ½ Æ ½ by choosing ½, provided that «is sufficiently small, and ½ is sufficiently large. Additionally taking into account (II.4), we hence obtain ½ «Å «ÏÀÉ Óµ Å ½ Å ½ «½ ½ µ ½ ½ µ Æ ½ Æ ½ (II.18) with ½ being a universal constant. The next step is to use the positivity of, as a quadratic form. This gives, cf. (II.12), ½ Î ½ µ µ µ (II.19)

27 BM-April 22, and similarly Î ½ µ ½ µ µ (II.20) The distinction between and becomes important now. Namely, inserting the commutator µ µ ½, using the second intertwining relation µ Àµ in (I.54), and taking into account that ½,, we have (cf. (I.92)) Î µ µ µ µ µ µ Àµ Àµ Àµ Àµ (II.21) Inserting (II.19) and (II.21) into the definition (II.11) of Å, we obtain that Å Î µ Î Àµ Àµ µ µ µ ½ µ (II.22) Ó where ÑÜ ÒÈ ½ is a universal constant, as proved in Lemma II.7 below. For all ½, Eqs. (II.22) and (I.79) yield Å ½ µ Å ½ µ ½µ ÀÉ µ Æ (II.23) Here we used the following lower bound, cf. (I.8), (I.15) and (I.16): For any

28 BM-April 22, , À µ Å «Å Æ ««provided «µ. This establishes (II.23), for all ½. Finally, we insert (II.23) into (II.18) and obtain (II.24) Å «ÏÀÉ Óµ (II.25) ½µ ½ ÀÉ µ ½ ½ ½ «½ Æ ½ µ ½µ ÀÉ µ ½ ½ ½ «½ µ Æ ½ for some universal constant ½. We choose ½ ½ ½ «½ (II.26) and observe that ½, since ½. Moreover, this choice of insures that the terms in brackets in (II.25) vanish. Therefore, Å «ÏÀÉ Óµ ½µ ½ ÀÉ µ½ ½µ ½ ½ «½µ ÀÉ µ½ ½µ ÀÉ µ½ ½ ½ ÀÉ µ½ (II.27) finishing the proof. II.1 Estimate on Ï Ó«Óµ in Terms of Âa µ We recall from (I.84) the definition of Ï Ó«Óµ, Ï Ó«Ó Üµ «Óµ Óµ Û Üµ Û µ Å (II.28)

29 BM-April 22, and the definition of Õ. We decompose Ï Ó«Óµ Ï Ó«Óµ Ï Ó«Ó µ, where Ï Ó«Ó Üµ Ï Ó«Ó Üµ Ï Ó«Ó µ (II.29) «Óµ Óµ Û Üµ Û µ Å Ï Ó«Ó µ «Óµ Óµ ½ Below, we prove the following estimate. Û µ Å Lemma II.2. For some universal constant ½ and all ½,  µ ½ Ï Ó«Óµ  µ (II.30) ½ «(II.31) where  is defined in Eqs. (II.6) and (II.8), and µ µ µ. Proof: We use a variant of the Cauchy-Schwarz inequality. For Ü Ê, we introduce the characteristic functions ½Ü Ê and ½Ü Ê, so that ½ (II.32) Note that if Ü Ê then ½ and Ü µ Ê, by (I.31). Furthermore, Ü µ ÑÒ½ Ê, for Ü Ê, by (I.32). Hence, for any, and Ü µ Ü µ Ê Ü µ (II.33) ½ Ü µ Ü µ ½ Ü µ ½ Ü µ ½ Ü µ ½ (II.34) where ½ ½ Ê µ ½ ÑÒ½ Ê ½ and ½ ÑÒ½ Ê ½ ½ ÑÒ½ Ê ½. Additionally using that and ½, we obtain Û Üµ Û (II.35) Û Üµ µ Û Û µ Û Üµ Û µ Û Üµ µ Û Üµ µ Û Û Û ½ ½ Û ½ ½ ½

30 BM-April 22, for some universal constant ½. Next, the triangle inequality implies that Óµ Óµ µ, and hence Óµ Óµ µ, uniformly in the reference point Ó. Thus, Eq. (II.35) implies that Óµ Óµ Û Üµ Û µ µ ½ ½ Now, let À and observe that and µ ܵ ܵ ½ µ ܵ Ò µ Ò Ó ½ µ ܵ µ µ ܵ ½ Ó ½ µ ܵ µ ½ ½ ܵ ܵ Ò µ Ó ½ µ ܵ Ò µ Ó ½ µ ܵ where µ µ. Eqs. (II.35) (II.38) and ½ imply that Å ÏÓ«Óµ ««(II.36) (II.37) (II.38)  µ ½  µ ½ (II.39) and hence (II.31). Next we turn to estimating Ï Ó«Ó µ which is a constant matrix. Here we use the fact that Ï Ó«Ó µ to bound Ï Ó«Ó µ by a term of order. Lemma II.3. For all ½, ½ Æ where Æ È ½ Å. Ï Ó«Ó µ Æ ½ «½ µ (II.40)

31 BM-April 22, Proof: We pick and apply the Cauchy-Schwarz inequality. Additionally using the triangle inequality, implying Óµ Óµ µ, and taking (I.15) and the symmetry into account, we obtain ÏÓ«Ó µ ««Ò ÑÜ µ µ ½ (II.41) ÓÒ ½ and hence (II.40) follows if we show that, for all, To this end, we observe that Ö µ ½ Ó ½ Ò µ ½ ½ ½ Ö Ö, which implies Ó ½ (II.42) µ µ (II.43) ½ µ µ µ µ ½ µ ½ ½ ÑÜ ½ using (I.16). This gives (II.42). II.2 Estimate on Ï Ó«Óµ Ï in Terms of Âa µ We begin by deriving two simple estimates which are of key importance to our approach. Lemma II.4. For some universal constant ½, all, and all Ü Ê, (II.44) (II.45) Proof: First, Eqs. (II.44) and (II.45) hold true trivially, for Ü Á ½, because on that set ÕÑÜ. Conversely, if Ü Á ½ then we distinguish the cases Ü Á and Ü Á. In the former case, ½, and there is nothing to prove. It remains to demonstrate Eqs. (II.44) and (II.45) for Ü Á ½ and Ü Á, i.e., for Ü Ê ½ and Ü Ê. For this, we remark that Eqs. (I.31) and (I.32) imply Ü Ê ½ µ ÑÒ½ Ê Ü µ (II.46)

32 BM-April 22, which completes the proof upon the choice ½ Ê ½ ÑÒ½ Ê ½. Lemma II.4 is used to estimate Ï Üµ ܵ À µ Å (II.47) in terms of  µ, where  is defined in Eqs. (II.6) and (II.8). Recall furthermore that µ µ µ, and that ÜÔÕ and À ÈÕ. We have the following estimates. Lemma II.5. For some universal constant ½, all «½, and all Ü Ê, Ü µ ܵ À µ  µ ܵ (II.48) Ü µ ܵ À µ  µ ܵ (II.49) Proof: We only give the proof of Eq. (II.48), the proof of Eq. (II.48) is similar. Furthermore we note that µ and it hence suffices to prove Eq. (II.48) for. First, we estimate in the region Ü Ê, where we have ½. This, Eq. (I.28), Eq. (I.30), and À ܵ «Ü µ µ µ give À µ ܵ µ µ Û «µ Ò Û Ü Ü µ Û µ µ Û «Û ½ «µ (II.50) Û Ü Ü µ Û µ Ó yielding Eq. (II.48) in the region Ü Ê.

33 BM-April 22, Conversely, on Ü Ê, we have ܵ ܵ Ü µ À µ µ µ Û «µ ½ ½ Ò µ µ ½ Ó ½ Û «Ü µ ½ µ Û ½ «µ (II.51) where we use (I.30) again and, which we established in Lemma II.4. Observing that on Ü Ê, we have ÑÒ½ Ê, and additionally using µ, we further obtain Ü µ ½ µ Ü µ (II.52) for the universal constant ½ ½ µ ÑÒ½ Ê ½ ½. Inserting (II.52) into (II.51), we arrive at Eq. (II.48) in the region Ü Ê, as well. Combining Lemma II.2 and Eq. (II.48) in Lemma II.5, we arrive at the following intermediate conclusion, Theorem II.6. For some universal constant ½ and all ½,  µ ½ Ï Ó«Óµ Ï Â µ ½ (II.53) where  is defined in Eqs. (II.6) and (II.8). II.3 Estimate on Âa µ in Terms of Ãb In this subsection we replace and in  by and, respectively. It turns out that it is convenient to slighty increase the weights µ µ by passing to µ defined by ½ «½ ƽ «½ «Æ ½µ Æ (II.54)

34 BM-April 22, where «µ ½ and ½µ Æ denotes the Æ Ø power of ½µ as a matrix with ½, i.e., µ Æ Ò ½ Ò Æ ½ Ò½µ Ò½ ÒÆ ½µ ÒÆ ½ (II.55) recalling that µ µ. To prove the convergence of the series in Eq. (II.54) we view ½µ as a real, symmetric (and thus selfadjoint) matrix È on equipped with the norm induced by the standard scalar product ³ ³. The operator norm of ½µ is bounded by, and hence ½, provided «½ that trivially «µ ½ which is equivalent to «½. We note µ ½ «µ (II.56) by ignoring all terms in the series (II.55) of order Æ. The following Proposition shows that possesses similar summability properties as µ and has the additional property (II.58), which µ does not. Lemma II.7. For «½, the weights µ defined in (II.54) exist, are symmetric,, vanish on the diagonal,, and obey the summability condition ÑÜ ½ «(II.57) Moreover they additionally obey the estimate «½ «Ñ µ Ñ (II.58) for all. Ñ Proof: We first apply the triangle inequality and (I.16) to derive ÑÜ ÑÜ ½µ Æ Ò ½ Ò Æ ½ Ò ½µ Ò½ Ò Æ ½µ ÒÆ ½ Æ (II.59) which yields (II.57), ÑÜ µ ½ «½ ƽ «½ «Æ ½ «(II.60)

35 BM-April 22, Next, setting «½ «µ, we note that ½. Hence ½ ½µ is invertible, and its inverse can be expanded in a norm-convergent Neumann series, ½ ½µµ ½ ½ Æ ½µ Æ ½ ½ ƽ ½µ Æ (II.61) and hence we have ½ ½µµ ½ ½ «(II.62) For the derivation of (II.58), we multiply (II.62) by ½ ½µ and obtain ½ ½ ½µ ««½µ (II.63) Since the matrix elements of are nonnegative, this implies ½µ (II.64) which is equivalent to (II.58). Next we use the new weights µ to implement the change from to. The main result of this subsection is a form bound on  µ in terms of Ã, where  and à are defined in Eqs. (II.6), (II.8), and (II.9), and we recall that µ µ µ. Theorem II.8. There exists a universal constant ½ ½ such that, for all ««½ and all ½,  µ  ½ ««Ã (II.65) Proof: We first remark that (II.65) is equivalent to  µ  ½ ««Ã (II.66) Secondly, the first inequality in (II.66) trivially follows from µ. Hence it remains to prove the second inequality in (II.66), and for this it is sufficient to show the existence of a universal constant ½ such that  à ««Â (II.67) To this end, we proceed as in the proof of Lemma II.5 and observe that (I.33) implies  à (II.68) «Ñ Ñ Ñ «Ñ Ñ Ñ

36 BM-April 22, Now we apply Lemma II.4 and (II.58),  à «Ñ ½ µ ÑÑ Ñ «Ñ ½ µ ««Ñ «½ µ ½ µ ««Ñ ½ µ Ñ Ñ Ñ Ñ Ñ «««Ñ Ñ Ñ Ñ (II.69) for some universal constant ½, where we additionally use,, «½, and ½ to derive the last inequality. This clearly implies (II.67). II.4 Estimate on Ãb in Terms of Äb The purpose of this section is to replace and in Ãb by µ ½ and µ ½, respectively, where is a small parameter which we later choose to be of order ½. This replacement requires the introduction of Ä, defined in Eqs. (II.7) and (II.10). The main result of this subsection is the following theorem. Theorem II.9. There exists a universal constant ½ ½ such that, for all ««½ and all ½, à ½ Ä Æ (II.70) where Æ È ½ Å is the number operator on À. Proof: We start with the observation that if Ø Ê then ص and ص are bounded below by a universal, positive constant ½. Conversely, if Ø Ê then ص ص ½. Thus we have the following estimates, Ü µ ½ Ü Ê ½ Ü Ê Ü µ ½ Ü µ Ü µ (II.71) Ü µ ½ Ü Ê ½ Ü Ê Ü µ ½ Ü µ Ü µ (II.72)

37 BM-April 22, and, as an immediate consequence, ½ ½ ½ ½ (II.73) (II.74) Inserting (II.73) (II.74) into the definition (II.9) of à and observing the definition (II.8) of Â, we obtain à ½ Ä Ò ½ Ä ½ µ ½ Ä ½ µ Ó Ò Â ½ ««Ã (II.75) where the last inequality results from Theorem II.8, which ensures the existence of a universal constant ½ such that  ½ ««µ Ã, provided ««is sufficiently small. Thus, if ««and µ ½ then we can solve in Eq. (II.75) for à and obtain Ó Ã ½ Ä ½ µ (II.76) proving the assertion. II.5 Estimate on Äb in Terms of Åb In this subsection we show that subtracting ½ from µ in Ä does not cause a significant error. This is our last preparatory step for the derivation of the main form bound (II.1). Again, we first make the form of the desired upper bound precise by using the operator Å, defined in Eq. (II.7) and Eqs. (II.11) (II.12). The following theorem states the main result of this subsection. Theorem II.10. There exists a universal constant ½ ½ such that, for all, Ä ½ Å Æ (II.77) where Æ È ½ Å is the number operator on À.

38 BM-April 22, Proof: First we note that the difference between Å and Ä is bounded by ½ Å Ä Üµ ½ ½ À µ ܵ ܵ À µ À µ (II.78) ܵ À µ Due to Eqs. (I.8) and (I.15), we have that À µ À µ «µ ½ µ Next, it follows from Eqs. (II.48) (II.49) and (I.30) that ܵ À µ «(II.79) (II.80) and, for Ò, ܵ À µ «(II.81) «µ for some universal constant ½, since. Inserting these estimates into the last term on the right side of (II.78), we obtain ܵ À µ «½ µ ½ µ ܵ À µ «µ µ «(II.82)

39 BM-April 22, where we make use of (II.56), (II.58), and assumed ««½ to be sufficiently small to derive the last inequality. Observe that the last line in (II.82) is bounded by a multiple of Â. Thus, putting together (II.78), (II.79), and (II.82), we have Å Ä Â (II.83) for some universal constant ½. Applications of Theorem II.8 and of Theorem II.9, choosing to be a small, but universal constant, yield Â Ã Ä ½ (II.84) for some universal constants ½. Thus, if ½ is sufficiently large, we can solve in Eq. (II.82) for Ä and get Ä ½ Å (II.85) for some universal constant ½.

40 BM-April 22, III Semiclassical Localization of the Partition Function This section is devoted to the proof of Theorem I.6. The method we use here is inspired by a related method which appeared first in a paper by Sjöstrand [19]. The method was used to study the first eigenvalue of a Schrödinger operator divided by the dimension. He showed that this object is in the semiclassical limit given, up to an exponentially small error, by using the first eigenvalue of the operator localized to an Ð ½ neighbourhood of a global minimum. The exponential error is uniform in dimension (i.e., universal). This method was later applied in [10] to Laplace integrals (partition functions) and in [16] to Laplace integrals and transfer operators. The results mentioned above are for localization in an Ð ½ neighbourhood, whereas our result is for localization in a slab of the form Ü Ü Æ, where and Æ. The two methods apply to both localization problems, but under different sets of conditions. The method presented here seems particularly well adapted for the model considered in this paper. We note that the «given by Theorem I.6 is not uniform in. In the related results mentioned above, «can be chosen uniformly in the size of the neighbourhood. III.1 The Dilated Interaction Throughout this section we impose Hypotheses 1 and 2. We prove Theorem I.6 through a sequence of lemmata. Lemma III.1. Let µ ½µ and ½ ÑÜ. There exists a universal constant, independent of, such that ÒÛ Ü Ü for all Ü Ê. Proof: We abbreviate Û ½ µü ½ µü Ó (III.1) ½ Ò Ü µ ½ µü Ó Ý ½ µ Ü and Þ Øµ ØÜ ½ ØµÝ (III.2)

41 BM-April 22, The Fundamental Theorem of Calculus yields, cf. also the estimate (B.2), ½ Û Ü Ü µ Û Ý Ý µ Û Þ Øµ Þ Øµµ Ø Ø ½ Ò Ü Ó Ý µ Û Þ Øµ Þ Øµ Ü Ý µ Û Þ Øµ Þ Øµ ½ Ü Ü Þ Øµµ Þ Øµµ Ø (III.3) Using that Þ Øµµ Ò Ü µ Þ Øµµ and Þ Øµµ ½ we rewrite the estimate above as follows, Û Ü Ü µ Û Ý Ý µ Ü Ü ½ ½ Ü Ø Þ Øµµ ½ Ü Ü Ü Ü µ Ý µ Ü Hence, by the symmetry in and, we get ½ Ü Ü Ø Ü ½ Þ Øµµ Ø Ø Þ Øµµ, Ø Ü µ Ý µ Ü (III.4) Û Ü Ü µ Û Ý Ý µ (III.5) Ü µ Ý µ Now we temporarily fix and decompose the configuration space Ê into the two disjoint regions Ò Ü Ê Ü Ý Ó and For Ü Ý ½ µü, we have Ò Ó Ü Ê Ü Ý (III.6) Ü Ü ½ µ (III.7) where we used that ½. Conversely, if Ü Ý ½ µü then

42 BM-April 22, (I.29) implies Ü Ü Ü µ Ý µ Ü ÑÜ Ü ÑÜ Ý Ü ½ Ò Ü µ Ý Ò µ Ó Ò Ü µ Ü Ò µ Ó Ò Ü µ Ü Ò Ü µø Ø ½ Ü µ Ý µ (III.8) Combining the estimates (III.7) (III.8) and inserting them into (III.5) yields the lemma. We now specify in terms of the metric on the lattice and a fixed reference point Ó. Lemma III.2. Fix Ó and Ó ½, and set Ó Æ Ó ½ Æ Ó µ Óµ. Then there exists a universal constant such that, for all Ü Ê, Ò Û Ü Ü µ Û ½ µü ½ µü µó ½ Ó µ Ò Ü Ó ½ µü Ø (III.9) Proof: In view of Lemma III.1 and the fact that ½ Ó, it is sufficient to show that ÑÜ ½ (III.10) for some universal constant ½. For Ó, we observe that the triangle inequality for the metric and Eq. (I.16) implies ½ Óµ Ó ½ Ó ½ Óµ Óµ ½ Ó ÑÜ µ Conversely, for Ó, we estimate Ó ½ Ó Óµ ½ Ó ½ Ó Óµ (III.11) Ó (III.12) using (I.16). This completes the proof since ½.