Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schrödinger operator Hε=-ε2Δε+Vε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H_\varepsilon = -\varepsilon ^2\Delta _\varepsilon + V_\varepsilon $$\end{document} on ℓ2(εZd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2(\varepsilon \mathbb {Z}^d)$$\end{document}, where ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is a small parameter and the potential Vε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_\varepsilon $$\end{document} is defined in terms of a multiwell energy landscape f on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}. It is unitarily equivalent to the generator of a diffusion on εZd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \mathbb {Z}^d$$\end{document}, satisfying the detailed balance condition with respect to the Boltzmann weight exp(-f/ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp {(-f/\varepsilon )}$$\end{document}. These type of diffusions exhibit metastable behavior and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of Hε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\varepsilon $$\end{document} in the semiclassical regime ε≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll 1$$\end{document} and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of f. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring–Kramers formula for the metastable tunneling time of the underlying stochastic process.


Introduction
This paper derives sharp semiclassical spectral asymptotics for Schrödinger operators acting on ℓ 2 pεZ d q of the form where ∆ ε is the discrete nearest-neighbor Laplacian of εZ d and V ε is a possibly unbounded multiplication operator, defined in terms of a multiwell energy landscape f .More precisely, given f P C 2 pR d q, we identify V ε with the function V ε pxq " e f pxq 2ε pε 2 ∆ ε e ´f 2ε qpxq.
(2) We shall dub H ε the discrete semiclassical Witten Laplacian associated with f .This is motivated by the following observation: the continuous space version of H ε , i.e. the Schrödinger operator H ε on L 2 pR d q obtained from (1), (2) by substituting ∆ ε with the Laplacian ∆ of R d , reads and thus coincides with the restriction on functions of the Witten Laplacian of R d [44,29,26,38].It is well known that the latter has deep connections to problems in Statistical Mechanics [23].In some situations, e.g. when considering lattice models of Statistical Mechanics as discussed below, one is led in a natural way to its discrete version (1), (2).The continuous space operator (3) is then rather a simplifying idealization of (1), (2): it is indeed easier to analyze H ε by exploiting the standard machinery of differential and semiclassical calculus, but the results might be a priori less accurate in making predictions.This paper shows a general strategy which permits to obtain sharp semiclassical estimates directly in the discrete setting.
We are mainly inspired by the analysis [25] on the continuous space Witten Laplacian and by the series of papers [32,33,34,35] by M. Klein and E. Rosenberger, who develop an approach to the semiclassical spectral analysis of discrete Schrödinger operators of the form (1) via microlocalization techniques.We refer also to the earlier work [30] and to [13,12] for semiclassical investigations in discrete settings.

Brief description of the main results.
Following in particular the approach of [33] we show that under mild regularity assumptions on f there is a low-lying spectrum of exponentially small eigenvalues which is well separated from the rest of the spectrum.Moreover the number of exponentially small eigenvalues equals the number of local minima of f , see Theorem 2.2 below.
Then we analyze in more detail the case of two local minima of f and compute the precise asymptotic splitting between the two small eigenvalues.From a general point of view, this corresponds to a subtle tunneling calculation through other, non-resonant wells [28] of the Schrödinger potential V ε , corresponding to saddle points of f .As opposed to [25] we work again under mild regularity assumptions on f and proceed with a streamlined, direct strategy that avoids WKB expansions, a priori Agmon estimates and also the underlying complex structure of the Witten Laplacian.Much of the simplification is obtained via a suitable choice of global quasimodes.We show that the leading asymptotic of the exponentially small eigenvalue gap is given by an Eyring-Kramers formula: λpεq " εAe ´E ε p1 `op1qq, where A, E ą 0 are explicit constants depending on f (see Theorem 2.3 for a precise statement) that turn out to coincide with the one obtained in the continuous case for H ε in [25] (see also [9,20]).In other terms, the geometric constraint imposed by the lattice turns out to be negligible in first order approximation.The vanishing rate of the remainder term depends on the regularity of f around its critical points.We show that f P C 3 pR d q implies an error of order Op ?εq.The spectral Eyring-Kramers formula in the discrete setting considered here is not new.Indeed, up to some minor variants, this type of result has been derived in the framework of discrete metastable diffusions, by analyzing mean transition times of Markov processes via potential theory [7].We shall discuss below more in detail the probabilistic interpretation of our results.The present paper shows that, as in the continuous setting, also in the discrete setting the Eyring-Kramers formula can be obtained by a direct and self-contained spectral approach, without relying at all on probabilistic potential theory.
We remark that the method we use to analyze the exponentially small eigenvalues can be extended also to the general case with more than two local minima.The extension is based on an iterative finite-dimensional matrix procedure, very similar to the one considered in [25] (see also [15] and references therein).This procedure is independent of the rest and not related to the peculiar analytical difficulties arising from the discrete character of the setting.To not obscure the exposition of the main ideas of this paper, the general case will be discussed somewhere else.
Connection to discrete metastable diffusions.
Our main motivation for investigating the spectral properties of H ε stems from its close connection to certain metastable diffusions with state space εZ d .These have been extensively studied in the probabilistic literature, mainly due to their paradigmatic properties and their applications to problems in Statistical Mechanics [11,7,4,19,2,37,40].The general, continuous time version might be described in terms of a Markovian generator L ε of the form L ε ψpxq " ÿ vPZ d r ε px, x `εvq rψpx `εvq ´ψpxqs , with r ε px, x `εvq being the rate of a jump from x to x `εv.The jump rates are assumed to satisfy the detailed balance condition with respect to the Boltzmann weight ρ ε " e ´f {ε on εZ d , so that L ε may be realized as a selfadjoint operator acting on the weighted space ℓ 2 pεZ d ; ρ ε q.Moreover the scaling is chosen so that L ε formally converges for ε Ñ 0 to a first order differential operator on R d , corresponding to a deterministic transport along a vector field.One might thus think of the dynamics as a small stochastic perturbation of a deterministic motion.A standard choice of jump rates satisfying the above requirements is given by r ε px, x `εvq " where pe 1 , . . ., e d q is the standard basis of R d .
There is a direct link between the discrete Witten Laplacian and discrete diffusions as described above: up to a change of sign and multiplicative factor ε, the Markovian generator L ε given by ( 4), (5) and the discrete Witten Laplacian given by ( 1),(2) are formally unitarily equivalent.This can be seen by the well-known ground state transformation, which turns a Schrödinger operator into a diffusion operator [31], see Proposition 2.5 below for the precise statement.As a consequence, our spectral analysis of H ε can be immediately translated into analogous results on L ε , see Corollary 2.6.The advantage of working with H ε is that in the flat space ℓ 2 pεZ d q one can exploit Fourier analysis and related microlocalization techniques.
We remark that discrete diffusions as described above naturally arise in the context of disordered mean field models in Statistical Mechanics.A prominent example is the dynamical random field Curie-Weiss model [21,7,4,40], which is well described by a discrete diffusion on εZ d after a suitable reduction in terms of order parameters.The limit ε Ñ 0 then corresponds to the thermodynamic limit of infinite volume.
A characteristic feature of the dynamics B t ψ " L ε ψ for small ε is metastability: if f admits several local minima the system remains trapped for exponentially large times in neighborhoods of local minima of f before exploring the whole state space.This is due to the fact that the local minima of f turn out to be exactly the stable equilibrium points of the limiting deterministic motion.We refer to [22,39,6] for comprehensive introductions to metastability of Markov processes and e.g. to [3,16,36] for shorter surveys.
A key issue in the understanding of metastability is to quantify the time scales at which metastable transitions between local minima occur.For discrete diffusions of type (4) sharp asymptotic estimates have been obtained in [8,7] in terms of average hitting times.The formula for the leading asymptotics is called Eyring-Kramers formula.In [7] it is also shown that there is a very clean relationshp between the metastable transition times and the low-lying spectrum of ´Lε .Indeed, there is a cluster of exponentially small eigenvalues, each one being asymptotically equivalent to the inverse of a metastable transition time.
The problem of determining the asymptotic behavior of metastable transition times can therefore be equivalently phrased as a problem of spectral asymptotics of the generator L ε and thus of H ε .Due to these facts, one can view the method presented in this paper as a spectral approach to the computation of metastable transition times in discrete setting.
Plan of the paper.In Section 2 we introduce the setting, provide precise definitions and basic properties for the discrete Witten Laplacian H ε , the diffusion generator L ε and state our main results: Theorem 2.2, saying that there are as many exponentially small eigenvalues of H ε as minima of f and that there is a large gap of order ε between them and the rest of the spectrum; Theorem 2.3, giving the precise splitting between exponentially small eigenvalues due to the tunnel effect (Eyring-Kramers formula).In Section 3 we collect some preliminary tools which can be seen as general means for a semiclassical analysis on the lattice: the IMS formula for the discrete Laplacian which permits to localize quadratic forms on the lattice; estimates on the discrete semiclassical Harmonic oscillator based on microlocalization techniques; and results on sharp Laplace asymptotics on the lattice εZ d based on the Poisson summation formula.In Section 4 and Section 5 we provide the proofs of Theorem 2.2 and Theorem 2.3 respectively.

Precise setting and main results
Throughout the paper we shall use the following notation.We consider the symmetric set N " te k , ´ek : k " 1, . . ., du Ă Z d , where pe 1 , . . ., e d q is the standard basis of R d .For ε ą 0 the symbols ∇ ε and ∆ ε denote respectively the rescaled discrete gradient and the rescaled discrete Laplacian of the lattice εZ d , with graph structure induced by εN .More precisely, for every ψ : εZ d Ñ R we define ∇ ε ψ px, vq " ε ´1 rψpx `εvq ´ψpxqs , @x P εZ d and v P N , We shall work on the Hilbert space ℓ 2 pεZ d q " tψ P R εZ d : }ϕ} ℓ 2 pεZ d q ă 8u, where } ¨}ℓ 2 pεZ d q is the norm corresponding to the scalar product The discrete Laplacian ∆ ε is a bounded linear operator on ℓ 2 pεZ d q.It is also selfadjoint and ´∆ε is nonnegative.More precisely, for ψ, ψ 1 P ℓ 2 pεZ d q, once can check that and in particular x´∆ ε ψ, ψy ℓ 2 pεZ d q " }∇ ε ψ} 2 ℓ 2 pεZ d ;R N q ě 0.Here } ¨}ℓ 2 pεZ d ;R N q is the norm induced by the scalar product defined for α, α 1 P ℓ 2 pεZ d ; R N q :" tα P R εZ d ˆN : }αp¨, vq} ℓ 2 pεZ d q ă 8 forall v P N u (the space of square integrable 1-forms on the graph εZ d ).
2.1.Definition and basic properties of H ε .Given a function f : R d Ñ R and a parameter ε ą 0, we define a new function V ε : R d Ñ R by setting Note that the expression (6) for V ε and the one given in the introduction in (2) are equal by definition of ∆ ε and ∇ ε .We shall identify in the sequel V ε with the corresponding multiplication operator in ℓ 2 pεZ d q having dense domain DompV ε q " tψ P ℓ 2 pεZ d q : V ε ψ P ℓ 2 pεZ d qu.The restriction of V ε to C c pεZ d q (i.e. the set of ψ P R εZ d such that ψpxq " 0 for all but finitely many x) is essentially selfadjoint.
We are interested in the Schrödinger-type operator H ε : DompV ε q Ñ ℓ 2 pεZ d q given by Note that H ε is a selfadjoint operator in ℓ 2 pεZ d q and its restriction to C c pεZ d q is essentially selfadjoint.This follows e.g. from the Kato-Rellich Theorem [43,Theorem 6.4], using the analogous properties of V ε and the fact that ∆ ε is bounded and selfadjoint.Moreover, from the pointwise bound V ε ě ´2d and the nonnegativity of ´∆ε it follows immediately that H ε is bounded from below.An important observation is that the quadratic form associated with H ε is not only bounded from below, but even nonnegative.This is due to the special form of the potential V ε .Indeed, a straightforward computation yields where ∇ f,ε denotes a suitably weighted discrete gradient: It follows in particular that the spectrum of H ε is contained in r0, 8q.
Remark 2.1.The property (7) states that H ε is the Laplacian associated to the distorted gradient ∇ f,ε .As it is done for the continuous space Witten Laplacian [44,29], it is possible to give an extension of H ε in the sense of Hodge theory.The extended operator is then defined on a suitable algebra of discrete differential forms and satisfies the usual intertwining relations.We shall not use this fact and refer to [15] for details.

Assumptions and main results.
We shall consider the following two sets of hypotheses on the function f .Here and in the following | ¨| denotes the standard euclidean norm on R d .The gradient and Hessian of a function on R d are denoted by ∇ and Hess.
(ii) Hess f is bounded on R d .
Note that H1 implies that the set of critical points of f is finite.Indeed, nondegenerate critical points are necessarily isolated and by (i) the critical points of f must be contained in a compact subset of R d .
To analyze the exponential splitting between small eigenvalues we will assume for simplicity the following more restrictive hypothesis.
H2. Hyptohesis H1 holds true.Moreover The first result we present shows that under Assumption H1 the essential spectrum of H ε , denoted by Spec ess pH ε q, is uniformly bounded away from zero and that its discrete spectrum, denoted by Spec disc pH ε q, is well separated into two parts: one consists of exponentially small eigenvalues, the other of eigenvalues which are at least at distance of order ε from zero.Moreover the rank of the spectral projector corresponding to the exponentially small eigenvalues equals exactly the number of local minima of f : Theorem 2.2.Assume H1 and denote by N 0 P N 0 the number of local minima of f .There exist constants ε 0 P p0, 1q and C ą 0 such that for each ε P p0, ε 0 s the following properties hold true.
(iii) H ε admits at least N 0 eigenvalues counting multiplicity.In the nontrivial case that N 0 ‰ 0, the N 0 -th eigenvalue λ N 0 pεq (according to increasing order and counting multiplicity) satisfies the bounds The properties stated in Theorem 2.2 are well-known in the continous space setting [41,27] and have also been recently extended to certain infinitedimensional situations [10].In the finite-dimensional continuous space setting the standard proof consists in approximating the Schrödinger operator with harmonic oscillators around the critical points of f .The error is then estimated using the IMS localization formula, which permits to connect the local estimates around the critical points to global estimates.The discrete case is analytically more difficult, due to the nonlocal character of the discrete Laplacian.The main idea to overcome these difficulties is taken from [33] and consists in localizing not only the potential V ε but the full operator H ε .This amounts in localizing the symbol in phase space and is also referred to as micolocalization.The setting in [33] is very general and requires the machinery of pseudodifferential operators, which makes the proof rather involved and requires strong regularity assumptions on the potential V ε which are not assumed here.Here we give a more elementary proof which is adapted to our special case and works well under Hypthesis H1.
This implies that e ´f 2ε is in the domain of H ε and therefore, since H ε e ´f 2ε " 0 by direct computation, that 0 is an eigenvalue of H ε .Moreover, due to the fact that N generates the group Z d , it follows for example from (7) that only multiples of Ψ ε can be eigenfunctions corresponding to the eigenvalue 0. Thus we conclude that 0 is an eigenvalue with multiplicity 1 for every ε ą 0. Since, by assumption, there are N 0 " 2 local minima of f , it follows from Theorem 2.2 that, for ε ą 0 sufficiently small, there is exactly one eigenvalue λ ε of H ε , which is different from 0 and is exponentially small in ε.Moreover, by the same theorem, λ ε must have multiplicity 1.Our second main result provides the precise leading asymptotic behavior of λpεq.This behavior is expressed in terms of two constants A, E ą 0, giving respectively the prefactor and the exponential rate.More precisely one defines where h ˚:" mintf pm 0 q, f pm 1 qu P R is the lowest energy level and where h ˚P R is given by the height of the barrier which separates the two minima.More precisely, h ˚can be defined as follows [25].For h P R we denote by S f phq :" f ´1 pp´8, hqq the (open) sublevel set of f corresponding to the height h and by N f phq the number of connected components of S f phq.Then h ˚pf q P R is defined as the maximal height which disconnects S f phq into two components: By simple topological arguments, on the level set f ´1ph ˚q there must be at least one critical point of f of index 1 and at most a finite number n of them, which we label in an arbitrary order as s 1 , . . ., s n .We denote by µps k q the only negative eigenvalue of Hess f ps k q.The constant A is then defined in terms of the quadratic curvature of f around the two minima and the relevant saddle points.More precisely, one defines , if f pm 0 q " f pm 1 q.
(10) Our second main theorem is the following.
Theorem 2.3.Assume H2 and take ε 0 ą 0 as in Theorem 2.2.Let A, E be given respectively by (8), (10) and let, for ε P p0, ε 0 q, λpεq be the smallest non-zero eigenvalue of H ε .Then the error term Rpεq, defined for ε P p0, ε 0 q by λpεq " εAe ´E ε p1 `Rpεqq , satisfies the following: there exists a constant C ą 0 such that |Rpεq| ď C ? ε for every ε P p0, ε 0 q.Remark 2.4.Stronger smoothness properties of f (f P C 4 pR d q should suffice) may lead to the improved bound R ε " Opεq.A possible proof may be obtained using the underlying Witten complex structure as explained in the author's PhD thesis [15].There it is shown that f P C 8 pR d q implies that R ε admits full asymptotic expansions in powers of ε.But the proof is substantially more involved, since it requires a construction and detailed analysis of discrete WKB expansions on the level of 1-forms.
As anticipated in the introduction, our main results can be easily translated into results on spectral properties of the class of metastable discrete diffusions with generator (4), (5).Since this might be a particularly interesting application of our results, we shall spell out precisely their consequences from the stochastic point of view.

Results on the diffusion operator L ε .
Given a function f : R d Ñ R and a parameter ε ą 0, we consider the weight functions ρ ε pxq " e ´f pxq ε and r ε px, x 1 q " 1 ε e ´f px 1 q´f pxq 2ε , @x, Note that ρ ε and r ε are related by the identity We work now in the weighted Hilbert space ℓ 2 pρ ε q obtained as subspace of R εZ d by introducing the weighted scalar product and the corresponding induced norm } ¨}ℓ 2 pρεq .We shall denote by L ε the Laplacian of the weighted graph εZ d , whose vertices are weighted by ρ ε and whose edges (determined by N ) are weighted by ρ ε r ε .More precisely we define L ε : DompL ε q Ñ ℓ 2 pρ ε q by setting DompL ε q " and, for each x P εZ d , This provides a Hilbert space realization of the formal operator (4),(5).
Proof.Let ε ą 0. We consider the unitary operator Then a direct computation shows that and that Φ ε rDompL ε qs " DompV ε q.
From the unitarily equivalence it follows that L ε is not only symmetric and nonnegative (this can be checked by summation by parts and using the detailed balance condition (11)), but also selfadjoint.We remark also that C c pεZ d q, which is a core for H ε and is invariant under Φ ε , is also a core of L ε .Combining Proposition 2.5 with Theorem 2.2 and Theorem 2.3 yields then the following result.
Corollary 2.6.Assume H1 and denote by N 0 P N 0 the number of local minima of f .There exist constants ε 0 P p0, 1q, C ą 0 such that for each ε P p0, ε 0 s the following properties hold true.(i) Spec ess p´L ε q Ă rε ´1C, 8q and | Spec disc p´L ε q X r0, Cs| ď N 0 .
(ii) ´Lε admits at least N 0 eigenvalues counting multiplicity.In the nontrivial case that N 0 ‰ 0, the N 0 -th eigenvalue λ N 0 pεq (according to increasing order and counting multiplicity) satisfies the bounds Moreover, assuming in addition H2, and taking A, E as in (8), (10), the error term Rpεq, defined for ε P p0, ε 0 q by satisfies the following: there exists a constant C ą 0 such that |Rpεq| ď C ? ε for every ε P p0, ε 0 q.
We stress that (i) implies a quantitative scale separation between the N 0 slow modes, corresponding to the metastable tunneling times, and all the other modes, corresponding to fast relaxations to local equilibria.In principle it is also possible to refine the analysis of the fast modes revealing the full hierarchy of scales governing the dynamics in the small ε regime, see [18] for the continuous space setting and a Γ-convergence formulation.
As already mentioned, the rigorous derivation of an Eyring-Kramers formula of type (13) in the setting of discrete metastable diffusions had already been derived by a different approach based on capacity estimates [8,6].Compared to these previous results the formula given in (13) differs in two aspects: 1) The estimate on the error term Rpεq is improved by our approach, since in [6, Theorem 10.9 and 10.10], under the same regularity assumptions as considered here (f P C 3 pR d q) a logarithmic correction appears.More precisely our result improves the error estimate from Rpεq " Op a εrlog 1{εs 3 q to Rpεq " Op ?εq.
2) The prefactor A given in ( 13) differs from the one given in [8,6].This is due to our slightly different choice of jump rates, compare (5) with [6, (10.1.2.), p. 248].Indeed it is clear that the prefactor is sensible to the particular choice of jump rates among the infinitely many possible jump rates satisfying the detailed balance condition with respect to the Boltzmann weight e ´f {ε .This sensitivity of the prefactor is opposed to the robustness of the exponential rate E, which is universal as can be seen e.g.via a Large Deviations analysis.We remark that, while the rates chosen in [6] correspond to a Metropolis algorithm, our choice (5) corresponds, in the context of the Statistical Mechanics models mentioned above, to a heat bath algorithm.This is a very natural choice and is considered for example in [37].As observed in the introduction, it is the choice which in first order approximation gives the same prefactor as the continuous space model (3).Furthermore, [8,6] concerns discrete time processes, which means that the rates are normalized and thus bounded over R d .Our setting includes also the case of possibly unbounded rates which requires some additional technical work for the analysis outside compact sets.

General tools for a semiclassical analysis on the lattice
This section is devoted to some preliminary tools for a semiclassical analysis on the lattice.Subsection 3.1 concerns a discrete IMS localization formula, see [43,Lemma 11.3] or [14,Theorem 3.2], where also an explanation of the name can be found, for the standard continuous space setting and [33].The IMS formula is a simple observation based on a computation of commutators.It will be used repeteadly for decomposing the quadratic form induced by a Schrödinger operator into localized parts.Subsection 3.2 provides estimates on the first two eigenvalues of the discrete semiclassical Harmonic oscillator.These estimates follow from more general results proven in [33].Nevertheless we shall include a relatively short and completely selfcontained proof, which focuses on the estimates needed to prove the separation between exponentially small eigenvalues of H ε and the rest of its spectrum, as provided by Theorem 2.2.The proof is based on a microlocalization which permits to separate high and low frequency actions of the operator.Subsection 3.3 provides sharp asymptotic results for Laplace-type sums.These are instrumental in almost all the computations necessary for deriving the Kramers formula for the eigenvalue splitting and for tunneling calculations in general.Our proofs are again based on Fourier analysis.In particular, following [15], we shall use the Poisson summation formula: shifting a function by an integer vector and summing over all shifts produces the same periodization as taking the Fourier series of the Fourier transform.Compared to [15], where it is shown how to get complete asymptotic expansions in the smooth setting, here we shall relax the regularity assumptions on the phase function to cover the applications we have in mind.

The discrete IMS formula.
We say that the set tχ j u jPJ is a smooth quadratic partition of unity of R d if J is a finite set, χ j P C 8 pR d q for every j P J and ř jPJ χ 2 j " 1. Proposition 3.1.There exists a constant C ą 0 such that for every ε ą 0, every ψ P ℓ 2 pεZ d q and every smooth quadratic partition of unity | Hess χ j pxq|}ψ} ℓ 2 pεZ d q .
Proof.We have Differentiating the relation ř j χ 2 j " 1 yields ř j χ j ∇χ j ¨v " 0 for every v and therefore, by Taylor expansion, for every x P R d and v P N , ˇˇˇˇ1 ´ÿ j The claim follows now from ( 14) and ( 15) by noting that the assumption ř j χ 2 j " 1 also implies sup j,x |χ j pxq| ď 1, that }ψp¨`εvq} ℓ 2 pεZ d q " }ψ} ℓ 2 pεZ d q for every v and recalling that N is bounded.

Estimates on the discrete semiclassical Harmonic oscillator.
We provide lower bounds for the first and the second eigenvalue of the semiclassical discrete Harmonic oscillator.Proposition 3.2.For every x P R d let U pxq " xx ´x, M px ´xqy, where x P R d and M is a symmetric d ˆd real matrix with strictly positive eigenvalues denoted by κ 1 , . . ., κ d .Moreover let λ 0 " ř j ?κ j and λ 1 " ř j ?κ j 2 min j ?κ j .Then there exist for every ε ą 0 a function Ψ ε P ℓ 2 pεZ d q and constants ε 0 , C ą 0 such that for every ε P p0, ε 0 s and ψ P C c pεZ d q the following hold: The proof is by localization around low frequencies in Fourier space and comparison with the corresponding continuous Harmonic oscillator on R d , whose first and second eigenvalue are given respectively by ελ and ελ 1 .At low frequencies, discrete and continuous Harmonic oscillators are close, while the high frequencies do not contribute to the bottom of the spectrum.
In the proof we shall use the following notation: for ε ą 0 and ψ P ℓ 1 pεZ d q we define ψpξq :" p2πq ´d 2 ÿ and for ε ą 0 and φ P L 1 pR d q we define φpξq :" Then by Parseval's theorem and by Plancherel's theorem We recall also the inversion theorem for the Fourier transform and Fourier series, which in our notation reads as follows.Let φ P SpR d q, the Schwartz space on R d and let φpxq " φp´xq for every x P R d .Then Moreover, for every φ P C 8 pR d q with supppφq Ă p´π, πq d it holds φ P ℓ 1 pεZ d q and φpξq " φpξq @ξ P r´π, Proof of Proposition 3.2 .Let ε P p0, 1s, ψ P C c pεZ d q and let ϕ :" `´ε 2 ∆ ε `U ˘ψ.
The four terms in the right hand side of ( 21) are analyzed separately in the following.
1) Analysis of the first term in the right hand side of (21).
Recalling that s " 2 5 and the Parseval theorem (16) we conclude that Final step.

3.3.
Laplace asymptotics on εZ d .Given x 0 P R d and δ ą 0 we denote by B δ px 0 q " tx P R d : |x ´x0 | ă δu the open ball of radius δ around x 0 and, for each ε ą 0, by B ε δ px 0 q " B δ px 0 qXεZ d its intersection with εZ d and by rB ε δ px 0 qs c " εZ d zB ε δ px 0 q the complementary of B ε δ px 0 q.Proposition 3.3.Let qpxq " 1 2 x ¨Qx, where Q is a symmetric, positive definite d ˆd matrix and let x 0 P R d and m P N 0 .Then there exists a γ ą 0 such that for every ε P p0, 1s Moreover for every δ ą 0 there exists γpδq ą 0 such that for every ε P p0, 1s ÿ Remark 3.4.The Gaussian integrals appearing on the right hand side of (30) can be computed explicitly.We shall use in the sequel the explicit value only for m " 0, in which case (30) becomes We shall also use the following estimate for odd moments: The latter follows from Proposition 3. It follows that Since û is a linear combination of derivatives of Gaussian functions, there exist constants C, γ ą 0 such that |ûpxq| ď Ce ´2γ|x| 2 @x P R d .
It follows that for every ε P p0, 1s which concludes the proof of (30).
In order to prove (31), fix δ ą 0 and note that, due to the positive definiteness of Q, there exists a constant C ą 0 such that qpxq ą Cδ 2 for every x P rB ε δ px 0 qs c .Thus, for ε P p0, 1s, ÿ with K " e Cδ 2 `şR d e ´qpx´x 0 q dx `1˘.
To see the last inequality one can use e.g. the Poisson summation formula for ε d ř xPεZ d e ´qpx´x 0 q .From (33), chosing γ ą 0 sufficiently small and C 1 ą 0 sufficiently large we obtain ÿ The estimate (31) for m " 0 follows then using (30) with m " 0. The case of positive m can be proven in the same way.
The following proposition concerns more general, not necessarily quadratic phase functions.
Moreover let m P N 0 .Then for ε P p0, 1s it holds where qpxq " 1 2 Hess ϕpx 0 qx ¨x for all x P R d .Remark 3.6.Under the stronger regularity assumption ϕ P C 8 pB δ px 0 qq one can show that the error term in (35) admits a complete asymptotic expansion in powers of ε, see [15, Appendix C] for details.
Note that, a fortiori, also ϕpxq ě qpxq for every x P B δ 1 px 0 q.Moreover, since ϕpxq |x| 2 is continuous and stricly positive on the compact set B δ px 0 qzB δ 1 px 0 q we can take e.g.α " mintα 1 , inf It will be enough to prove (35) with the sum on the left hand side restricted to B ε δ 1 px 0 q, since by Proposition 3.3 there exists a γ ą 0 s.t. for ε P p0, 1s ÿ We shall consider the decomposition with, setting for short u ε pxq " |x ´x0 | 2m e ´qpxq ε , ´e´r pxq ε ´1 ´ε´1 rpxq ¯uε pxq.
It follows from Proposition 3.3 that there exists a γ ą 0 s.t. for ε P p0, 1s Morever, using |rpxq| ď C|x ´x0 | 3 for all x P B δ px 0 q and (36) gives with the last estimate being a consequence of Proposition 3.3.Finally, in order to analyze the term I 1 pεq, we consider first the case k " 3. We then have by ( 32) which together with (37), ( 38) and (39) finishes the proof for k " 3.For the case k " 4 we write rpxq " t 3 pxq `ρpxq, where t 3 : B δ 1 px 0 q Ñ R is the cubic term in the Taylor expansion of ϕ around x 0 , thus satisfying t 3 px 0 `xq " t 3 px 0 ´xq, and ρ : B δ 1 px 0 q Ñ R satisfies |ρpxq| ď C 1 |x ´x0 | 4 for some C 1 ą 0. We then have and therefore by Proposition 3.3 which finishes the proof in the case k " 4.

Proof of Theorem 2.2
Recall the definition of V ε given in (6).To prove Theorem 2.2 we shall reduce to suitable localized problems and then exploit basic pointwise estimates on V ε as stated in the following two complementary lemmata.The first one gives a uniform strictly positive lower bound on V ε away from critical points.
The second one concerns the local behavior of V ε around critical points.
Note that these bounds are almost immediate to obtain, even under weaker assumptions, if instead of V ε one considers the corresponding continuous space potential 1 4 |∇f | 2 ´ε 2 ∆f appearing in (3).The discrete case follows from straightforward Taylor expansions and elementary estimates.We shall give the details of the arguments at the end of this section for completeness.Lemma 4.1.Assume f P C 2 pR d q and that Hess f is bounded on R d .Let S Ă R d and a ą 0 such that |∇f pxq| ą a for every x P S. Then there exist constants ε 0 , C ą 0 such that V ε pxq ě C @x P S and @ε P p0, ε 0 s.
Then there exists a constant C ą 0 such that for all x P B R pzq and ε ą 0 After these preliminary estimates on V ε we turn to the proof of Theorem 2.2.We first show that the essential spectrum of H ε is bounded from below by a constant, as claimed in Theorem 2.2 (i).
Remark 4.4.The proof given below shows that the claim of Proposition 4.3 still holds without assuming that the critical points of f are nondegenerate.Also the regularity assumption on f can be relaxed by assuming f P C 2 pR d q instead of f P C 3 pR d q.
Proof.Let χ :" α1 K , where 1 K is the indicator function of a bounded set K Ă R d and α P R. Then χ, seen as a multiplication operator in ℓ 2 pεZ d q, is of finite rank (in particular compact) for every ε ą 0. It follows from Weyl's theorem that for fixed ε ą 0, inf Spec ess `Hε ˘" inf Spec ess `Hε `χ˘.
The claim follows by chosing α and K large enough so that for some constants ε 0 , C ą 0 the inequality V ε pxq `χpxq ě C holds for every x P R d and ε P p0, ε 0 s.To see that this choice is possible recall the uniform bound V ε ě ´2d and note that by Assumption H1 (i) there exist a ą 0, R ą 0 such that |∇f pxq| ą a for |x| ą R. It follows then by Lemma 4.1 that for suitable C, ε 0 ą 0 it holds V ε pxq ě C for |x| ą R and ε P p0, ε 0 q.
The next proposition provides the crucial estimate for the proof of statement (ii) in Theorem 2.2.
Proof of Corollary 4.6.By Proposition 4.3 and Proposition 4.5 we can find ε 0 , C ą 0 such that Spec ess pH ε q Ă rCε 0 , 8q @ε P p0, ε 0 s (42) and such that (41) holds.If for every ε P p0, ε 0 q it happens that | Spec disc pH ε q X r0, Cε 2 s| ď N 0 , the claim is proven.Thus, we only have to check the case in which there exists ε ˚P p0, ε 0 q such that | Spec disc pH ε˚q X r0, Cε2 s| ą N 0 . ( But this case is impossible.Indeed (43) implies that there exist at least N 0 `1 distinct eigenvalues of H ε˚i n r0, Cε2 s and thus in particular the N 0 `1-th eigenvalue λ N 0 `1pε ˚q(in increasing order and counting multiplicity) exists and satisfies In particular λ N 0 `1pε ˚q ď Cε 0 2 and therefore, by (42), λ N 0 `1pε ˚q is smaller than the bottom of the essential spectrum.From this, the Max-Min principle and ( 41) it follows that where the infimum is taken over all normalized ψ P V K ε X DompV ε q, with V ε being the linear span of the set tΨ 1,ε , . . ., Ψ N 0 ,ε u Ă ℓ 2 pεZ d q appearing in (41).But ( 44) and ( 45) are in contradiction.
1) Analysis of the first term in the right hand side of (46).
We apply Proposition 3.2: let κ 1 pz j q . . ., κ d pz j q be the eigenvalues of 1 2 Hess f pz j q, so that in particular 1 2 ∆f pz j q " ř i κ i pz j q and κ 2 1 pz j q . . ., κ 2 d pz j q are the eigenvalues of 1  4 rHess f pz j qs 2 .
Using ř N j"0 χ 2 j,ε " 1 gives Since there is a constant K ą 0 such that sup xPR d | Hess χ j,ε pxq| ď Kε ´2s for every ε P p0, εs and j " 0, . . ., N , it follows from Lemma 3.1 that there exists a constant C 4 ą 0 such that In particular we shall use that Final step.
It remains to show part (iii) of Theorem 2.2 to complete the proof.In order to do so, we can assume N 0 ‰ 0, since otherwise there is nothing to prove.By the Max-Min principle [24, Theorem 11.7 and Proposition 11.9] together with the bound on the essential spectrum given by Proposition (4.3) it is sufficient to show that for each ε ą 0 there exist N 0 orthonormal functions in the domain DompV ε q of H ε such that the quadratic form associated with H ε is exponentially small for each of these functions.We shall now exhibit such a family of orthonormal functions.
Let tz 1 , . . ., z N 0 u be the set of local minima of f .We fix δ ą 0 such that B 3δ pz k q X B 3δ pz j q is the empty set for k ‰ j and such that f ą f pz k q on B 3δ pz k qztz k u.Moreover we fix for each k " 1, . . ., N 0 a cutoff function χ k P C 8 pR d ; r0, 1sq, satisfying χ " 1 on B δ pz k q χ " 0 on R d zB 2δ pz k q.We consider then for each ε ą 0 and for each k " 1, . . ., N 0 the functions ψ k,ε : R d Ñ R given by Then for each ε ą 0 the (restrictions to εZ d of the) functions ψ 1,ε , . . ., ψ N 0 ,ε are in the domain of H ε and orthonormal in ℓ 2 pεZ d q.Moreover the following proposition shows that the quadratic form associated with H ε is exponentially small for each of these functions and thus concludes the proof of Theorem 2.2.
In the remainder of this section we provide the proofs of the basic estimates on V ε given in Lemma 4.1 and Lemma (4.2).
Proof of Lemma 4.1.A Taylor expansion gives for every x P R d the representation where, thanks to the boundedness of Hess f , DR ą 0 s.t.|R ε px, vq| ď R @x P R d , v P N and ε P p0, 1s.
In fact, one may write Using that cosh t´1 ě t 2 with t " ∇f pxq¨v 2 and ř vPN |∇f pxq¨v| 2 " 2|∇f pxq| 2 we get for ε P p0, mint1, 1 R uq and every x P R d the lower bound In particular x P S and @ε P p0, mint1, 1 R uq.
Proof of Lemma 4.2.This follows from a straightforward Taylor expansion.Indeed, fixing z P R d such that ∇f pzq " 0 and R ą 0, we have on B R pzq the uniform estimate ´1 2ε rf p¨`εvq ´f s " ´1 2 ∇f ¨v ´ε 4 Hess f v ¨v `Opε 2 q.
Using the inequality |e t ´1 ´t| ď 1 2 t 2 e |t| with t " ε 4 Hess f v ¨v `Opε 2 q then gives The expansion cosh x " 1 `1 2 x 2 `Opx 4 q and the equalities Expanding all terms in x around z, which gives in particular |∇f pxq| 2 " rHess f pzqs 2 px ´zq ¨px ´zq `Op|x ´z| 3 q and ∆f pxq " ∆f pzq `Op|x ´z|q, finishes the proof.

General strategy.
In order to compute the precise asymptotics of the smallest non-zero eigenvalue λpεq of H ε we shall consider a suitable choice of an ε-dependent test function ψ ε .The latter will be referred to as quasimode and its precise construction will be given in Subsection 5.2.Since ψ ε will be chosen orthogonal to the ground state e ´f {p2εq for every ε, the upper bound on λpεq given in Theorem 2.3 will follow immediately from the Max-Min principle, giving and from the precise computation of the right hand side in the above formula by using the Laplace asymptotics on εZ d given in Subsection 3.3.The result of these computations is the content of Proposition 5.2 and Proposition 5.3.
The proof of the lower bound on λpεq given in Theorem 2.3 is more subtle.We shall derive it as a corollary of Theorem 2.2 and the following abstract estimate, which was used in [17] in a similar way.
Proposition 5.1.Let pT, DpT qq be a nonnegative selfadjoint operator on a Hilbert space pX, x¨, ¨yq.Moreover let τ ą 0 and P " 1 r0,τ s pT q be the spectral projector of T corresponding to the interval r0, τ s and let λ " suppr0, τ s X SpecpT qq.Then for every normalized u P DpT q with xT u, uy ‰ 0 it holds λ ě xT u, uy p1 ´Rpuqq , where Rpuq ě 0 satisfies rRpuqs 2 " τ ´1 xT u, T uy xT u, uy .
Proof.We denote by } ¨} the Hilbert space norm and fix a u P DpT q such that }u} " 1 and xT u, uy ‰ 0. Then we have the estimates The latter estimate will be a consequence of Proposition 5.3 and Proposition 5.4, which is proven again by analyzing the Laplace asymptotics of a sum over εZ d .
We shall assume throughout the rest of this section that the Assumption H2 is satisfied.

Definition of the quasimode ψ ε .
Let s 1 , . . ., s n be the relevant saddle points of f , i.e. the critical points of index one of f appearing in formula (10) defining the prefactor A. Given x P R d we associate to it a linear "reaction coordinate" ξ k " ξ k pxq around the saddle point s k , which parametrizes the unstable direction of Hess f ps k q.
More precisely, we chose one of the two normalized eigenvectors corresponding to the only negative eigenvalue µps k q of Hess f ps k q, denote it by τ k , and set ξ k pxq " xx ´sk , τ k y @k " 1, . . ., n.
(61) Recalling our notation S f phq " f ´1 pp´8, hqq for the open sublevel set of f corresponding to the height h P R, we consider for ρ ą 0 and k " 1, . . ., n the closed set R k " and the open set B " S f ph ˚`ρqz p Ť k R k q.Henceforth the parameter ρ ą 0 appearing in the definition of R k and B is fixed sufficiently small such that the following properties hold: -the set B has exactly two connected components B p0q and B p1q , containing respectively m 0 and m -For each k " 1, . . ., n the function ϕ k :" f `|µps k q|ξ 2 k satisfies ϕ k pxq ą f ps k q for every x P R k zts k u.
Note that Hess ϕ k ps k q " | Hess f ps k q|.In other terms the quadratic approximation of ϕ k around s k is obtained from that of f by flipping the sign of the only negative eigenvalue of Hess f ps k q.
Let ε P p0, 1s.The quasimode ψ ε for the spectral gap is defined as follows.We define first on the sublevel set S f ph ˚`ρq and χ P C 8 pR; r0, 1sq satisfies χ " 1 on r´ρ 3 , ρ 3 s, χpηq " 0 for |η| ě 2 3 ρ and χpηq " χp´ηq.Note that Note also that for each k " 1, . . .n the of the vector τ k defining ξ k (see (61)) can be chosen such that κ ε is C 8 on S f ph ˚`ρq, which we shall assume in the sequel.In order to extend κ to a smooth function defined on the whole R d we introduce another cutoff function θ P C 8 pR d ; r0, 1sq by setting for x P R d θpxq " # 1 for x P S f ph ˚`ρ 2 q 0 for x P R d zS f ph ˚`3 4 ρq .
Finally we define the quasimode ψ ε by setting for xθκε,e ´f {ε y ℓ 2 pεZ d q Note that ψ ε P C 8 pR d q with compact support.In particular its restiction to εZ d , which we still denote by ψ ε , is in C c pεZ d q Ă DompV ε q.Moreover, it follows from its very definition that ψ ε is orthogonal to the ground state e ´f {p2εq with respect to the scalar product x¨, ¨yℓ 2 pεZ d q .
We now state the crucial estimates concerning the quasimode ψ ε .The proofs follow from straightforward computations exploiting the results of Subsection 3.3 on the Laplace asymptotics for sums over εZ d .We shall give the details in Subsection 5.4.
Proposition 5.2.Assume H2 and let ε P p0, 1s.The function ψ ε defined in (63) satisfies where h ˚" mintf pm 0 q, f pm 1 qu is the minimum of f and pdet Hess f pm 0 qq ´1 2 if f pm 0 q ă f pm 1 q.
Proposition 5.3.Assume H2 and let ε P p0, 1s.The function ψ ε defined in (63) satisfies where µps k q is the only negative eigenvalue of Hess f ps k q and h ˚is defined in (9).
Remark 5.5.With the stronger assumption f P C 4 pR d q the Op ?εq error terms appearing in Proposition 5.2 and Proposition 5.2 can be shown to be actually Opεq.Indeed it is enough to apply Proposition 3.5 with k " 4 instead of k " 3 each time it is used in the proofs given below.

Proofs of the quasimode estimates.
Proof of Proposition 5.2.Let ε P p0, 1s.We first consider the case f pm 0 q ă f pm 1 q.Then there exist α, δ ą 0 such that f ě f pm 0 q `α on rB δ pm 0 qs c and θκ ε " ´1 on B δ pm 0 q.It follows that, denoting for short by Ω δ ε the bounded set rB ε δ pm 0 qs c X supppθq, it holds e ´f pxq{ε ´1 `Ope ´rf pm 0 q`αs{ε q ¯.Proposition 3.5 gives then The same arguments and the estimate |θκ ε | ď 1 show that xθκ ε , e ´f {ε y ℓ 2 pεZ d q " ´p2πεq d 2 pdet Hess f pm 0 qq ´1 2 e ´f pm 0 q `1 `Op ?εq ˘. (65) Taking the quotient between (64) and (65) it follows then from the definition of ψ ε that The norm ψ ε } 2 ℓ 2 pεZ d q can be now computed by splitting again the sum in two sums, respectively over B ε δ pm 0 q and Ω ε δ .The conclusion in the case f pm 0 q ă f pm 1 q follows by again using Proposition 3.5 for the first sum and arguing as above for the second sum.
We now consider the case f pm 0 q " f pm 1 q.It follows from the definition of Let α, δ ą 0 such that f ě f pm 0 q `α on rB δ pm 0 q Y B δ pm 1 qs c and θκ ε " ´1 on B δ pm 0 q, θκ ε " 1 on B δ pm 1 q.With arguments as above one gets xθκ, e ´f {ε y ℓ 2 pεZ d q " " p2πεq Putting these expressions into (66), the desired result (5.2) follows after some algebraic manipulations.
Proof of Proposition 5.3.Let ε P p0, 1s.Using (7) and the notation F ε px, vq " 1 2 rf pxq `f px `εvqs gives Since the function θ has support in S f ph ˚`3 4 ρq, we can restrict (for ε sufficiently small) the sum running over εZ d to the bounded set εZ d XS f ph ˚`ρq.Note that S f ph ˚`ρq is the union of the disjoint sets B and Ť k pR k zrS f ph ˚ρqs c q. 1 We write in the sequel for short R k,ε :" εZ d X pR k zrS f ph ˚`ρqs c q and B ε :" εZ d X B and discuss below separately the sum over Y n k"1 R k,ε , which will give the main contribution, and the sum over B ε , which will give a negligible contribution.
In order to get rid of θ we take δ ą 0 small enough such that for each k it holds B ε δ ps k q Ă R k,ε X S f ph ˚`1 4 ρq Ă R k,ε .Since θ and κ ε are uniformly bounded in ε and f ě h ˚`1 4 ρ on R k,ε zB ε δ ps k q, we get using (67) that, for ε ą 0 sufficiently small (and thus also for ε P p0, 1s), it holds rκ ε px `εvq ´κε pxqs 2 e ´Fεpx,vq{ε `Ope ´ph ˚`ρ 4 q q, (68 where we have used also that for ε sufficiently small θpxq " θpx `εvq " 1 for x P B ε δ ps k q.
We discuss now in detail the behavior of x Þ Ñ κ ε px `εvq ´κε pxq near s k .For k " 1, . . ., n and x P R 1 k , v P N and ε P p0, 1s consider the function G " G k,x,v,ε : r0, 1s Ñ R defined by where for shortness we have set ϕ k pxq " f pxq ´f ps k q `|µps k q|ξ 2 k pxq and α k pxq " pe j ¨τk q 2 cosh B j f pxq 2 `|µps k q|ξ k pxqpe j ¨τk q 3 sinh B j f pxq 2 * " 1 `Op|x ´sk | 2 q.
Putting together (68), (72), using Proposition 3.5, summing over k and the fact that f ps k q " h ˚for every k finally gives 2) Analysis on B ε .

As in
Step 1) we get rid of θ by considering the set B 1 ε " B ε XS f ph ˚`1 4 ρq Ă B. Arguing as before and now using that κ ε pxq " κ ε px `εvq for every x P B 1 ε , v P N and ε sufficiently small, gives then rκ ε px `εvq ´κε pxqs 2 e ´Fεpx,vq{ε `Ope ´ph ˚`ρ 4 q q " Ope ´ph ˚`ρ 4 q q.
Proof of Proposition 5.4.The isomporphism (12) gives the identity Since the function θ has support in S f ph ˚`3 4 ρq, we can restrict (for ε sufficiently small) the sum over εZ d to the bounded set εZ d X S f ph ˚`ρq.As in the proof of Proposition 5.3 we shall split the latter into the disjoint sets Y k R k,ε ,with R k,ε :" εZ d X pR k zrS f ph ˚`ρqs c q, and B ε :" εZ d X B.
We discuss here in detail only the contribution coming from the sets R k,ε .Indeed the sum over B ε can be neglected arguing exactly as in Step 2) of Proposition 5.3 and using instead of (67) that by Taylor expansion e ´1 2 ∇εf px,vq " e ´∇f pxq¨v{2 p1 `Opεqq . (73) As in the proof of Proposition 5.3 we first get rid of θ by taking a δ ą 0 small enough such that for each k it holds B ε δ ps k q Ă R k,ε X S f ph ˚`1 4 ρq Ă R k,ε .Since θ and κ ε are uniformly bounded in ε and f ě h ˚`1 4 ρ on R k,ε zB ε δ ps k q, we get using (73) that, for ε ą 0 sufficiently small (and thus also for ε P p0, 1s), it holds shows that the first order terms in (76) cancel out and thus α k pxq " Op|x śk | 2 q.It follows then from (74), (75) that there exists a constant C ą 0 such that for every ε P p0, 1s and every k " 1, . . ., n |x ´sk | 4 e ´ϕk pxq{ε " Opε 3 qe ´h˚{ ε , with ϕ k pxq " f pxq`|µps k q|ξ 2 k pxq´f ps k q and with the last estimate following from Proposition 3.5 by taking m " 2.
(i) lim inf |x|Ñ8 f pxq |x| ą 0. (ii) The function f has exactly two local minimum points m 0 , m 1 P R d .

ε d 4 ÿ
xPR k,ε ˜ÿ vPN e ´1 2 ∇εf px,vq ε∇ ε pθκ ε qpx, vq ¸2 e ´f pxq{ε ď Cε d`1 e ´f ps k q{ε ÿ xPB ε δ ps k q Hess f pxqv ¨v| ď ε 4 A|v| 2 , one gets for ε P p0, 1s and every x P R d |R ε px, vq| ď A|v| 2 4 e ε pxq " ÿ vPN " e x | ě }P u} 2 λ " xu, λP uy ě xu, T P uy "We shall apply Proposition 5.1 to the case T " H ε , τ " Cε, where C is the constant appearing in Theorem 2.2 and u " p}ψ ε } ℓ 2 pεZ d q q ´1ψ ε , where ψ ε is the same quasimode used for the upper bound on λpεq.By Theorem 2.2 (ii) we thus obtain a lower bound on λpεq.The fact that this lower bound coincides with the lower bound given in Theorem 2.3 is a consequence of the precise computation of the right hand side of (60), which we already mentioned (see Prop. 5.2 and Prop.5.3), and the estimate