aa r X i v : . [ m a t h - ph ] F e b SPECTRAL ANALYSIS OF DISCRETE METASTABLEDIFFUSIONS
GIACOMO DI GES `U
Abstract.
We consider a discrete Schr¨odinger operator H ε “ ´ ε ∆ ε ` V ε on ℓ p ε Z d q , where ε ą V ε isdefined in terms of a multiwell energy landscape f on R d . This operatorcan be seen as a discrete analog of the semiclassical Witten Laplacianof R d . It is unitarily equivalent to the generator of a diffusion on ε Z d ,satisfying the detailed balance condition with respect to the Boltzmannweight exp p´ f { ε q . These type of diffusions exhibit metastable behaviorand arise in the context of disordered mean field models in Statisti-cal Mechanics. We analyze the bottom of the spectrum of H ε in thesemiclassical regime ε ! f . Then we analyze in more detail the bistable case and compute theprecise asymptotic splitting between the two exponentially small eigen-values. Through this purely spectral-theoretical analysis of the discreteWitten Laplacian we recover in a self-contained way the Eyring-Kramersformula for the metastable tunneling time of the underlying stochasticprocess. Introduction
This paper derives sharp semiclassical spectral asymptotics for Schr¨odingeroperators acting on ℓ p ε Z d q of the form H ε “ ´ ε ∆ ε ` V ε , ă ε ! , (1)where ∆ ε is the discrete nearest-neighbor Laplacian of ε Z d and V ε is a pos-sibly unbounded multiplication operator, defined in terms of a multiwellenergy landscape f . More precisely, given f P C p R d q , we identify V ε withthe function V ε p x q “ e f p x q ε p ε ∆ ε e ´ f ε qp x q . (2)We shall dub H ε the discrete semiclassical Witten Laplacian associated with f . This is motivated by the following observation: the continuous space ver-sion of H ε , i.e. the Schr¨odinger operator H ε on L p R d q obtained from (1),(2)by substituting ∆ ε with the Laplacian ∆ of R d , reads H ε “ ´ ε ∆ ` | ∇ f | ´ ε ∆ f, (3)and thus coincides with the restriction on functions of the Witten Laplacianof R d [44, 29, 26, 38]. It is well known that the latter has deep connectionsto problems in Statistical Mechanics [23]. In some situations, e.g. when Mathematics Subject Classification.
Key words and phrases.
Metastability, Semiclassical spectral theory, Spectral gap, Wit-ten Laplacian, discrete Schr¨odinger Operators, Mean field models. G. DI GES `U considering lattice models of Statistical Mechanics as discussed below, oneis led in a natural way to its discrete version (1),(2). The continuous spaceoperator (3) is then rather a simplifying idealization of (1),(2): it is indeedeasier to analyze H ε by exploiting the standard machinery of differentialand semiclassical calculus, but the results might be a priori less accurate inmaking predictions. This paper shows a general strategy which permits toobtain sharp semiclassical estimates directly in the discrete setting.We are mainly inspired by the analysis [25] on the continuous space Wit-ten 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 analysisof discrete Schr¨odinger operators of the form (1) via microlocalization tech-niques. We refer also to the earlier work [30] and to [13, 12] for semiclassicalinvestigations in discrete settings. Brief description of the main results.
Following in particular the approach of [33] we show that under mild regu-larity assumptions on f there is a low-lying spectrum of exponentially smalleigenvalues which is well separated from the rest of the spectrum. Moreoverthe number of exponentially small eigenvalues equals the number of localminima of f , see Theorem 2.2 below.Then we analyze in more detail the case of two local minima of f andcompute the precise asymptotic splitting between the two small eigenvalues.From a general point of view, this corresponds to a subtle tunneling calcula-tion through other, non-resonant wells [28] of the Schr¨odinger 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 expan-sions, a priori Agmon estimates and also the underlying complex structure ofthe Witten Laplacian. Much of the simplification is obtained via a suitablechoice of global quasimodes. We show that the leading asymptotic of theexponentially small eigenvalue gap is given by an Eyring-Kramers formula: λ p ε q “ εAe ´ Eε p ` o p qq , where A, E ą f (see Theorem 2.3for a precise statement) that turn out to coincide with the one obtainedin the continuous case for H ε in [25] (see also [9, 20]). In other terms, thegeometric constraint imposed by the lattice turns out to be negligible in firstorder approximation. The vanishing rate of the remainder term depends onthe regularity of f around its critical points. We show that f P C p R d q implies an error of order O p? ε q .The spectral Eyring-Kramers formula in the discrete setting consideredhere is not new. Indeed, up to some minor variants, this type of result hasbeen derived in the framework of discrete metastable diffusions, by analyzing PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 3 mean transition times of Markov processes via potential theory [7]. We shalldiscuss below more in detail the probabilistic interpretation of our results.The present paper shows that, as in the continuous setting, also in thediscrete setting the Eyring-Kramers formula can be obtained by a directand self-contained spectral approach, without relying at all on probabilisticpotential theory.We remark that the method we use to analyze the exponentially smalleigenvalues can be extended also to the general case with more than twolocal minima. The extension is based on an iterative finite-dimensionalmatrix procedure, very similar to the one considered in [25] (see also [15] andreferences therein). This procedure is independent of the rest and not relatedto the peculiar analytical difficulties arising from the discrete character ofthe 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 stemsfrom its close connection to certain metastable diffusions with state space ε Z d . These have been extensively studied in the probabilistic literature,mainly due to their paradigmatic properties and their applications to prob-lems in Statistical Mechanics [11, 7, 4, 19, 2, 37, 40]. The general, continuoustime version might be described in terms of a Markovian generator L ε ofthe form L ε ψ p x q “ ÿ v P Z d r ε p x, x ` εv q r ψ p x ` εv q ´ ψ p x qs , (4)with r ε p x, x ` εv q being the rate of a jump from x to x ` εv . The jumprates are assumed to satisfy the detailed balance condition with respect tothe Boltzmann weight ρ ε “ e ´ f { ε on ε Z d , so that L ε may be realized as aselfadjoint operator acting on the weighted space ℓ p ε Z d ; ρ ε q . Moreover thescaling is chosen so that L ε formally converges for ε Ñ R d , corresponding to a deterministic transport alonga vector field. One might thus think of the dynamics as a small stochasticperturbation of a deterministic motion. A standard choice of jump ratessatisfying the above requirements is given by r ε p x, x ` εv q “ ε e ´ ε rp f p x ` εv q´ f p x qs if v P t´ e k , e k u k “ ,...,d , , (5)where p e , . . . , e d q is the standard basis of R d .There is a direct link between the discrete Witten Laplacian and discretediffusions as described above: up to a change of sign and multiplicativefactor ε , the Markovian generator L ε given by (4),(5) and the discrete WittenLaplacian given by (1),(2) are formally unitarily equivalent. This can be seenby the well-known ground state transformation, which turns a Schr¨odingeroperator into a diffusion operator [31], see Proposition 2.5 below for the G. DI GES `U precise statement. As a consequence, our spectral analysis of H ε can beimmediately translated into analogous results on L ε , see Corollary 2.6. Theadvantage of working with H ε is that in the flat space ℓ p ε Z d q one can exploitFourier analysis and related microlocalization techniques.We remark that discrete diffusions as described above naturally arise inthe context of disordered mean field models in Statistical Mechanics. Aprominent example is the dynamical random field Curie-Weiss model [21, 7,4, 40], which is well described by a discrete diffusion on ε Z d after a suitablereduction in terms of order parameters. The limit ε Ñ B t ψ “ L ε ψ for small ε is metasta-bility: if f admits several local minima the system remains trapped for expo-nentially large times in neighborhoods of local minima of f before exploringthe 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 deter-ministic motion. We refer to [22, 39, 6] for comprehensive introductions tometastability 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 timescales at which metastable transitions between local minima occur. Fordiscrete diffusions of type (4) sharp asymptotic estimates have been obtainedin [8, 7] in terms of average hitting times. The formula for the leadingasymptotics is called Eyring-Kramers formula. In [7] it is also shown thatthere is a very clean relationshp between the metastable transition times andthe low-lying spectrum of ´ L ε . Indeed, there is a cluster of exponentiallysmall eigenvalues, each one being asymptotically equivalent to the inverseof a metastable transition time.The problem of determining the asymptotic behavior of metastable tran-sition times can therefore be equivalently phrased as a problem of spectralasymptotics of the generator L ε and thus of H ε . Due to these facts, onecan view the method presented in this paper as a spectral approach to thecomputation of metastable transition times in discrete setting. Plan of the paper.
In Section 2 we introduce the setting, provide precisedefinitions and basic properties for the discrete Witten Laplacian H ε , thediffusion generator L ε and state our main results: Theorem 2.2, saying thatthere 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 thespectrum; Theorem 2.3, giving the precise splitting between exponentiallysmall eigenvalues due to the tunnel effect (Eyring-Kramers formula). In Sec-tion 3 we collect some preliminary tools which can be seen as general meansfor a semiclassical analysis on the lattice: the IMS formula for the discreteLaplacian which permits to localize quadratic forms on the lattice; estimateson the discrete semiclassical Harmonic oscillator based on microlocalizationtechniques; and results on sharp Laplace asymptotics on the lattice ε Z d based on the Poisson summation formula. In Section 4 and Section 5 weprovide the proofs of Theorem 2.2 and Theorem 2.3 respectively. PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 5 Precise setting and main results
Throughout the paper we shall use the following notation. We consider thesymmetric set N “ t e k , ´ e k : k “ , . . . , d u Ă Z d , where p e , . . . , e d q is the standard basis of R d . For ε ą ∇ ε and ∆ ε denote respectively the rescaled discrete gradient and the rescaleddiscrete Laplacian of the lattice ε Z d , with graph structure induced by ε N .More precisely, for every ψ : ε Z d Ñ R we define ∇ ε ψ p x, v q “ ε ´ r ψ p x ` εv q ´ ψ p x qs , @ x P ε Z d and v P N , ∆ ε ψ p x q “ ε ´ ÿ v P N r ψ p x ` εv q ´ ψ p x qs , @ x P ε Z d . We shall work on the Hilbert space ℓ p ε Z d q “ t ψ P R ε Z d : } ϕ } ℓ p ε Z d q ă 8u ,where } ¨ } ℓ p ε Z d q is the norm corresponding to the scalar product x ψ, ψ y ℓ p ε Z d q “ ε d ÿ x P ε Z d ψ p x q ψ p x q . The discrete Laplacian ∆ ε is a bounded linear operator on ℓ p ε Z d q . It isalso selfadjoint and ´ ∆ ε is nonnegative. More precisely, for ψ, ψ P ℓ p ε Z d q ,once can check that x´ ∆ ε ψ, ψ y ℓ p ε Z d q “ x ψ, ´ ∆ ε ψ y ℓ p ε Z d q “ x ∇ ε ψ, ∇ ε ψ y ℓ p ε Z d ; R N q , and in particular x´ ∆ ε ψ, ψ y ℓ p ε Z d q “ } ∇ ε ψ } ℓ p ε Z d ; R N q ě . Here } ¨ } ℓ p ε Z d ; R N q is the norm induced by the scalar product x α, α y ℓ p ε Z d ; R N q “ ε d ÿ x P ε Z d ÿ v P N α p x, v q α p x, v q , defined for α, α P ℓ p ε Z d ; R N q : “ t α P R ε Z d ˆ N : } α p¨ , v q} ℓ p ε Z d q ă 8 forall v P N u (the space of square integrable 1-forms on the graph ε Z 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 newfunction V ε : R d Ñ R by setting V ε p x q “ ÿ v P N “ e ´ ∇ ε f p x,v q ´ ‰ , @ x P R d . (6)Note that the expression (6) for V ε and the one given in the introductionin (2) are equal by definition of ∆ ε and ∇ ε . We shall identify in the sequel V ε with the corresponding multiplication operator in ℓ p ε Z d q having densedomain D om p V ε q “ t ψ P ℓ p ε Z d q : V ε ψ P ℓ p ε Z d qu . The restriction of V ε to C c p ε Z d q (i.e. the set of ψ P R ε Z d such that ψ p x q “ x ) is essentially selfadjoint. G. DI GES `U
We are interested in the Schr¨odinger-type operator H ε : D om p V ε q Ñ ℓ p ε Z d q given by H ε “ ´ ε ∆ ε ` V ε . Note that H ε is a selfadjoint operator in ℓ p ε Z d q and its restriction to C c p ε Z d q is essentially selfadjoint. This follows e.g. from the Kato-RellichTheorem [43, Theorem 6.4], using the analogous properties of V ε and thefact that ∆ ε is bounded and selfadjoint.Moreover, from the pointwise bound V ε ě ´ d and the nonnegativity of ´ ∆ ε it follows immediately that H ε is bounded from below. An impor-tant observation is that the quadratic form associated with H ε is not onlybounded from below, but even nonnegative. This is due to the special formof the potential V ε . Indeed, a straightforward computation yields x H ε ψ, ψ y ℓ p ε Z d q “ } ∇ f,ε ψ } ℓ p ε Z d ; R N q ě , @ ψ P D om p V ε q , (7)where ∇ f,ε denotes a suitably weighted discrete gradient: ∇ f,ε ψ p x, v q “ εe ´ f p x q` f p x ` εv q ε ∇ ε p e f ε ψ q p x, v q , @ x P ε Z d and v P N . It follows in particular that the spectrum of H ε is contained in r , . Remark 2.1.
The property (7) states that H ε is the Laplacian associatedto the distorted gradient ∇ f,ε . As it is done for the continuous space WittenLaplacian [44, 29] , it is possible to give an extension of H ε in the sense ofHodge theory.The extended operator is then defined on a suitable algebra ofdiscrete differential forms and satisfies the usual intertwining relations. Weshall 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. H1. f P C p R d q and all its critical points are nondegenerate. Moreover(i) lim inf | x |Ñ8 | ∇ f p x q| ą .(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 criticalpoints of f must be contained in a compact subset of R d .To analyze the exponential splitting between small eigenvalues we will as-sume for simplicity the following more restrictive hypothesis. H2.
Hyptohesis H1 holds true. Moreover(i) lim inf | x |Ñ8 f p x q| x | ą .(ii) The function f has exactly two local minimum points m , m P R d . PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 7
The first result we present shows that under Assumption H1 the essentialspectrum of H ε , denoted by Spec ess p H ε q , is uniformly bounded away fromzero and that its discrete spectrum, denoted by Spec disc p H ε q , is well sep-arated into two parts: one consists of exponentially small eigenvalues, theother of eigenvalues which are at least at distance of order ε from zero. More-over the rank of the spectral projector corresponding to the exponentiallysmall eigenvalues equals exactly the number of local minima of f : Theorem 2.2.
Assume H1 and denote by N P N the number of localminima of f . There exist constants ε P p , q and C ą such that for each ε P p , ε s the following properties hold true.(i) Spec ess p H ε q Ă r C, .(ii) | Spec disc p H ε q X r , Cε s| ď N .(iii) H ε admits at least N eigenvalues counting multiplicity. In the non-trivial case that N ‰ , the N -th eigenvalue λ N p ε q (according toincreasing order and counting multiplicity) satisfies the bounds ď λ N p ε q ď e ´ C { ε . The properties stated in Theorem 2.2 are well-known in the continous spacesetting [41, 27] and have also been recently extended to certain infinite-dimensional situations [10]. In the finite-dimensional continuous space set-ting the standard proof consists in approximating the Schr¨odinger operatorwith harmonic oscillators around the critical points of f . The error is thenestimated using the IMS localization formula, which permits to connectthe local estimates around the critical points to global estimates. The dis-crete case is analytically more difficult, due to the nonlocal character of thediscrete Laplacian. The main idea to overcome these difficulties is takenfrom [33] and consists in localizing not only the potential V ε but the full op-erator H ε . This amounts in localizing the symbol in phase space and is alsoreferred to as micolocalization. The setting in [33] is very general and re-quires the machinery of pseudodifferential operators, which makes the proofrather involved and requires strong regularity assumptions on the potential V ε which are not assumed here. Here we give a more elementary proof whichis adapted to our special case and works well under Hypthesis H1.We now assume the stronger Hypothesis H2. Then, thanks to the superlineargrowth condition H2 (i), it holds } e ´ f ε } ℓ p ε Z d q ă 8 , @ ε ą . This implies that e ´ f ε is in the domain of H ε and therefore, since H ε e ´ f ε “ H ε . Moreover, due to thefact that N generates the group Z d , it follows for example from (7) thatonly multiples of Ψ ε can be eigenfunctions corresponding to the eigenvalue0. Thus we conclude that 0 is an eigenvalue with multiplicity 1 for every ε ą N “ f , it followsfrom Theorem 2.2 that, for ε ą 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 G. DI GES `U main result provides the precise leading asymptotic behavior of λ p ε q . Thisbehavior is expressed in terms of two constants A, E ą
0, giving respectivelythe prefactor and the exponential rate. More precisely one defines E : “ h ˚ ´ h ˚ , (8)where h ˚ : “ min t f p m q , f p m 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 denoteby S f p h q : “ f ´ pp´8 , h qq the (open) sublevel set of f corresponding to theheight h and by N f p h q the number of connected components of S f p h q . Then h ˚ p f q P R is defined as the maximal height which disconnects S f p h q into twocomponents: h ˚ : “ max t h P R : N f p h q “ u . (9)By simple topological arguments, on the level set f ´ p h ˚ q there must beat 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 , . . . , s n . We denote by µ p s k q the only negative eigenvalue of Hess f p s k q . The constant A is thendefined in terms of the quadratic curvature of f around the two minima andthe relevant saddle points. More precisely, one defines A : “ $’&’%ř nk “ | µ p s k q| π p det Hess f p m qq | det Hess f p s k q| , if f p m q ă f p m q , ř nk “ | µ p s k q| π p det Hess f p m qq `p det Hess f p m qq | det Hess f p s k q| , if f p m q “ f p m q . (10)Our second main theorem is the following. Theorem 2.3.
Assume H2 and take ε ą as in Theorem 2.2. Let A, E be given respectively by (8) , (10) and let, for ε P p , ε q , λ p ε q be the smallestnon-zero eigenvalue of H ε . Then the error term R p ε q , defined for ε P p , ε q by λ p ε q “ εAe ´ Eε p ` R p ε qq , satisfies the following: there exists a constant C ą such that | R p ε q| ď C ? ε for every ε P p , ε q . Remark 2.4.
Stronger smoothness properties of f ( f P C p R d q should suf-fice) may lead to the improved bound R ε “ O p ε q . A possible proof may beobtained using the underlying Witten complex structure as explained in theauthor’s PhD thesis [15] . There it is shown that f P C p R d q implies that R ε admits full asymptotic expansions in powers of ε . But the proof is substan-tially more involved, since it requires a construction and detailed analysis ofdiscrete WKB expansions on the level of -forms. As anticipated in the introduction, our main results can be easily translatedinto results on spectral properties of the class of metastable discrete diffu-sions with generator (4), (5). Since this might be a particularly interestingapplication of our results, we shall spell out precisely their consequencesfrom the stochastic point of view.
PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 9
Results on the diffusion operator L ε . Given a function f : R d Ñ R and a parameter ε ą
0, we consider the weightfunctions ρ ε p x q “ e ´ f p x q ε and r ε p x, x q “ ε e ´ f p x q´ f p x q ε , @ x, x P R d . Note that ρ ε and r ε are related by the identity ρ ε p x q r ε p x, x q “ ρ ε p x q r ε p x , x q , @ ε ą x, x P R d . (11)We work now in the weighted Hilbert space ℓ p ρ ε q obtained as subspace of R ε Z d by introducing the weighted scalar product x ψ, ψ y ℓ p ρ ε q “ ε d ÿ x P ε Z d ψ p x q ψ p x q ρ ε p x q , and the corresponding induced norm } ¨ } ℓ p ρ ε q . We shall denote by L ε theLaplacian of the weighted graph ε Z d , whose vertices are weighted by ρ ε andwhose edges (determined by N ) are weighted by ρ ε r ε . More precisely wedefine L ε : D om p L ε q Ñ ℓ p ρ ε q by settingD om p L ε q “ ψ P ℓ p ρ ε q : ÿ v P N r ε p x, x ` εv q r ψ p x ` εv q ´ ψ p x qs P ℓ p ρ ε q + , and, for each x P ε Z d , L ε ψ p x q “ ÿ v P N r ε p x, x ` εv q r ψ p x ` εv q ´ ψ p x qs , @ ψ P D om p L ε q . This provides a Hilbert space realization of the formal operator (4),(5).
Proposition 2.5.
For each ε ą the operators ´ εL ε and H ε are unitarilyequivalent.Proof. Let ε ą
0. We consider the unitary operatorΦ ε : ℓ p ρ ε q Ñ ℓ p ε Z d q , Φ ε r ψ sp x q “ ? ρ ε p x q ψ p x q . Then a direct computation shows that H ε ψ “ ´ ε Φ ε “ L ε Φ ´ ε r ψ s ‰ , @ ψ P D om p V ε q , (12)and that Φ ε r D om p L ε qs “ D om p V ε q . (cid:3) From the unitarily equivalence it follows that L ε is not only symmetric andnonnegative (this can be checked by summation by parts and using thedetailed balance condition (11)), but also selfadjoint. We remark also that C c p ε Z 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 thenthe following result. Corollary 2.6.
Assume H1 and denote by N P N the number of localminima of f . There exist constants ε P p , q , C ą such that for each ε P p , ε s the following properties hold true.(i) Spec ess p´ L ε q Ă r ε ´ C, and | Spec disc p´ L ε q X r , C s| ď N . (ii) ´ L ε admits at least N eigenvalues counting multiplicity. In thenontrivial case that N ‰ , the N -th eigenvalue λ N p ε q (accordingto increasing order and counting multiplicity) satisfies the bounds ď λ N p ε q ď e ´ Cε . Moreover, assuming in addition H2, and taking
A, E as in (8) , (10) , theerror term R p ε q , defined for ε P p , ε q by λ p ε q “ Ae ´ Eε p ` R p ε qq , (13) satisfies the following: there exists a constant C ą such that | R p ε q| ď C ? ε for every ε P p , ε q . We stress that (i) implies a quantitative scale separation between the N slowmodes, corresponding to the metastable tunneling times, and all the othermodes, corresponding to fast relaxations to local equilibria. In principle itis also possible to refine the analysis of the fast modes revealing the fullhierarchy of scales governing the dynamics in the small ε regime, see [18] forthe continuous space setting and a Γ-convergence formulation.As already mentioned, the rigorous derivation of an Eyring-Kramers for-mula of type (13) in the setting of discrete metastable diffusions had alreadybeen derived by a different approach based on capacity estimates [8, 6].Compared to these previous results the formula given in (13) differs in twoaspects:1) The estimate on the error term R p ε q is improved by our approach,since in [6, Theorem 10.9 and 10.10], under the same regularity as-sumptions as considered here ( f P C p R d q ) a logarithmic correctionappears. More precisely our result improves the error estimate from R p ε q “ O p a ε r log 1 { ε s q to R p ε q “ O p? ε q .2) The prefactor A given in (13) differs from the one given in [8, 6]. Thisis 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 issensible to the particular choice of jump rates among the infinitelymany possible jump rates satisfying the detailed balance conditionwith respect to the Boltzmann weight e ´ f { ε . This sensitivity of theprefactor is opposed to the robustness of the exponential rate E ,which is universal as can be seen e.g. via a Large Deviations anal-ysis. We remark that, while the rates chosen in [6] correspond to aMetropolis algorithm, our choice (5) corresponds, in the context ofthe Statistical Mechanics models mentioned above, to a heat bathalgorithm. This is a very natural choice and is considered for exam-ple in [37]. As observed in the introduction, it is the choice which infirst order approximation gives the same prefactor as the continuousspace model (3). Furthermore, [8, 6] concerns discrete time pro-cesses, which means that the rates are normalized and thus boundedover R d . Our setting includes also the case of possibly unboundedrates which requires some additional technical work for the analysisoutside compact sets. PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 11 General tools for a semiclassical analysis on the lattice
This section is devoted to some preliminary tools for a semiclassical analysison the lattice.Subsection 3.1 concerns a discrete IMS localization formula, see [43, Lemma11.3] or [14, Theorem 3.2], where also an explanation of the name can befound, for the standard continuous space setting and [33]. The IMS for-mula is a simple observation based on a computation of commutators. Itwill be used repeteadly for decomposing the quadratic form induced by aSchr¨odinger operator into localized parts.Subsection 3.2 provides estimates on the first two eigenvalues of the dis-crete semiclassical Harmonic oscillator. These estimates follow from moregeneral results proven in [33]. Nevertheless we shall include a relatively shortand completely selfcontained proof, which focuses on the estimates neededto prove the separation between exponentially small eigenvalues of H ε andthe rest of its spectrum, as provided by Theorem 2.2. The proof is basedon a microlocalization which permits to separate high and low frequencyactions of the operator.Subsection 3.3 provides sharp asymptotic results for Laplace-type sums.These are instrumental in almost all the computations necessary for de-riving the Kramers formula for the eigenvalue splitting and for tunnelingcalculations 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 producesthe same periodization as taking the Fourier series of the Fourier transform.Compared to [15], where it is shown how to get complete asymptotic expan-sions in the smooth setting, here we shall relax the regularity assumptionson the phase function to cover the applications we have in mind.3.1. The discrete IMS formula.
We say that the set t χ j u j P J is a smooth quadratic partition of unity of R d if J is a finite set, χ j P C p R d q for every j P J and ř j P J χ j ” Proposition 3.1.
There exists a constant C ą such that for every ε ą ,every ψ P ℓ p ε Z d q and every smooth quadratic partition of unity t χ j u j P J itholds ››› ∆ ε ψ ´ ÿ j P J χ j ∆ ε p χ j ψ q ››› ℓ p ε Z d q ď C sup x,j | Hess χ j p x q|} ψ } ℓ p ε Z d q . Proof.
We have∆ ε ψ ´ ÿ j χ j ∆ ε p χ j ψ q p x q “ ε ÿ v P N « ´ ÿ j χ j p x q χ j p x ` εv q ff ψ p x ` εv q , thus ››› ∆ ε ψ ´ ÿ j χ j ∆ ε p χ j ψ q ››› ℓ p ε Z d q ď ε ÿ v P N sup x P R d ˇˇˇˇˇ ´ ÿ j χ j p x q χ j p x ` εv q ˇˇˇˇˇ } ψ p¨ ` εv q} ℓ p ε Z d q . (14) Differentiating the relation ř j χ j ” ř j χ j ∇ χ j ¨ v ” v and therefore, by Taylor expansion, for every x P R d and v P N , ˇˇˇˇˇ ´ ÿ j χ j p x q χ j p x ` εv q ˇˇˇˇˇ ď ε sup y P R d ÿ j | χ j p y q| | Hess χ j p y q v ¨ v | . (15)The claim follows now from (14) and (15) by noting that the assumption ř j χ j ” j,x | χ j p x q| ď
1, that } ψ p¨ ` εv q} ℓ p ε Z d q “ } ψ } ℓ p ε Z d q for every v and recalling that N is bounded. (cid:3) Estimates on the discrete semiclassical Harmonic oscillator.
We provide lower bounds for the first and the second eigenvalue of thesemiclassical discrete Harmonic oscillator.
Proposition 3.2.
For every x P R d let U p x q “ x x ´ ¯ x, M p x ´ ¯ x qy , where ¯ x P R d and M is a symmetric d ˆ d real matrix with strictly positive eigenvaluesdenoted by κ , . . . , κ d . Moreover let λ “ ř j ? κ j and λ “ ř j ? κ j ` j ? κ j . Then there exist for every ε ą a function Ψ ε P ℓ p ε Z d q andconstants ε , C ą such that for every ε P p , ε s and ψ P C c p ε Z d q thefollowing hold:(i) x ` ´ ε ∆ ε ` U ˘ ψ, ψ y ℓ p ε Z d q ě ε ´ λ ´ Cε ¯ } ψ } ℓ p ε Z d q . (ii) x ` ´ ε ∆ ε ` U ˘ ψ, ψ y ℓ p ε Z d q ě ε ´ λ ´ Cε ¯ } ψ } ℓ p ε Z d q ´x ψ, Ψ ε y ℓ p ε Z d q . The proof is by localization around low frequencies in Fourier space andcomparison with the corresponding continuous Harmonic oscillator on R d ,whose first and second eigenvalue are given respectively by ελ and ελ . Atlow frequencies, discrete and continuous Harmonic oscillators are close, whilethe high frequencies do not contribute to the bottom of the spectrum.In the proof we shall use the following notation: for ε ą ψ P ℓ p ε Z d q we define ˆ ψ p ξ q : “ p π q ´ d ÿ x P ε Z d ψ p x q e ´ i x ¨ ξε for ξ P R d , and for ε ą φ P L p R d q we defineˇ φ p ξ q : “ p π q ´ d ż R d φ p x q e i x ¨ ξε dx for ξ P R d . Then by Parseval’s theorem } ψ } ℓ p ε Z d q “ } ˆ ψ } L pr´ π,π s d q @ ψ P ℓ p ε Z d q , (16)and by Plancherel’s theorem } φ } L p R d q “ } ˇ φ } L p R d q @ φ P L p R d q X L p R d q . (17)We recall also the inversion theorem for the Fourier transform and Fourierseries, which in our notation reads as follows. Let φ P S p R d q , the Schwartzspace on R d and let ˜ φ p x q “ ˇ φ p´ x q for every x P R d . Then φ p ξ q “ ˇ˜ φ p ξ q @ ξ P R d . (18)Moreover, for every φ P C p R d q with supp p φ q Ă p´ π, π q d it holds ˇ φ P ℓ p ε Z d q and φ p ξ q “ ˆˇ φ p ξ q @ ξ P r´ π, π s d . (19) PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 13
Proof of Proposition 3.2 .
Let ε P p , s , ψ P C c p ε Z d q and let ϕ : “ ` ´ ε ∆ ε ` U ˘ ψ .Then φ P C c p ε Z d q and ˆ ϕ “ p W ´ A ε q ˆ ψ , where W : R d Ñ R is a multiplica-tion operator given by W p ξ q : “ d ÿ j “ sin ˆ ξ j ˙ , and A ε is a second order differential operator given by A ε : “ d ÿ j,k “ M j,k ` ε B j B k ` ε x k i B j ´ ¯ x k ¯ x j ˘ . It follows then by Parseval’s theorem (16) that x ` ´ ε ∆ ε ` U ˘ ψ, ψ y ℓ p ε Z d q “ xp W ´ A ε q ˆ ψ, ˆ ψ y L pr´ π,π s d q . (20)We now consider a cut-off function θ P C p R d ; r , sq which equals 1 on t ξ : | ξ | ď u and vanishes on t ξ : | ξ | ě u . For j “ , . . . , N we define with s “ the ε -dependent smooth quadratic partition of unity t θ ,ε , θ ,ε u bysetting θ ,ε p ξ q : “ θ ` ε ´ s p ξ q ˘ θ ,ε p ξ q : “ b ´ θ ,ε p ξ q . Moreover we denote by W the leading term in the ξ -expansion of the func-tion W around the origin, i.e. W p ξ q : “ Hess W p q ξ ¨ ξ “ | ξ | @ ξ P R d . A simple rearrangement of terms gives xp W ´ A ε q ˆ ψ, ˆ ψ y L pr´ π,π s d q “ xp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q `xp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q ` E p ε q ` E p ε q , (21)where the localization errors E p ε q , E p ε q are given by E p ε q : “ xp W ´ W q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q , E p ε q : “ ´ ÿ j “ xp θ j,ε A ε ´ A ε θ j,ε q ˆ ψ, θ j,ε ˆ ψ y L pr´ π,π s d q . The four terms in the right hand side of (21) are analyzed separately in thefollowing.
1) Analysis of the first term in the right hand side of (21) . Using that supp θ ,ε Ă p´ π, π q d for ε P p , s , Plancherel’s theorem (17) andthat the smallest eigenvalue of the Harmonic Oscillator ´ ε ∆ ` U on R d is λ , gives xp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q “ xp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L p R d q “x ` ´ ε ∆ ` U ˘ θ ,ε ˆ ψ, θ ,ε ˆ ψ y L p R d q ě ελ ››› θ ,ε ˆ ψ ››› L p R d q “ ελ } θ ,ε ˆ ψ } L p R d q “ ελ } θ ,ε ˆ ψ } L pr´ π,π s d q @ ε P p , s . (22) Moreover, considering for ε ą g ε p x q : “ e ´ x x, ? Mx y ε ›› e ´ x x, ? Mx y ε ›› L p R d q , an analogous computation gives for ε P p , s the estimate xp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q “ x ` ´ ε ∆ ` U ˘ θ ,ε ˆ ψ, θ ,ε ˆ ψ y L p R d q ě ελ x θ ,ε ˆ ψ, g ε y L p R d q ` ελ ˆ } θ ,ε ˆ ψ } L p R d q ´ x θ ,ε ˆ ψ, g ε y L p R d q ˙ “ ελ } θ ,ε ˆ ψ } L p R d q ´ ε min j ? κ j x θ ,ε ˆ ψ, g ε y L p R d q “ ελ } θ ,ε ˆ ψ } L pr´ π,π s d q ´ ε min j ? κ j x θ ,ε ˆ ψ, ˇ˜ g ε y L p R d q , where for the last equality the Fourier inversion theorem (18) is used for g ε .Moreover using (17), (19) and (16) we get x θ ,ε ˆ ψ, ˇ˜ g ε y L p R d q “ x θ ,ε ˆ ψ, ˜ g ε y L p R d q “ x ˆ ψ, θ ,ε ˜ g ε y L pr´ π,π s d q “x ˆ ψ, z~ θ ,ε ˜ g ε y L pr´ π,π s d q “ x ψ, ˇ θ ,ε ˜ g ε y L pr´ π,π s d q “ x ψ, ~ θ ,ε ˜ g ε y ℓ p ε Z d q . Thus, setting for shortnessΦ ε p ξ q : “ ? ε min j κ j ˜ g ε p ξ q for ξ P R d , we can conclude that xp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q ě (23) ελ } θ ,ε ˆ ψ } L pr´ π,π s d q ´ x ψ, θ ,ε Φ ε y ℓ p ε Z d q @ ε P p , s .
2) Analysis of the second term in the right hand side of (21) . Using the inequality sin t ě t for t P r , π s gives W p ξ q “ d ÿ j “ sin ˆ ξ j ˙ ě | ξ | @ ξ P r´ π, π s d . (24)Since supp θ ,ε Ă t ξ P R d : | ξ | ě ε u for ε P p , s , the bound (24) implies x W θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q ě ε } θ ,ε ˆ ψ } L pr´ π,π s d q @ ε P p , s . (25)Moreover, since ˆ ψ is periodic and θ ,ε equals 1 around the boundary of r´ π, π s d for ε P p , s , integration by parts gives x´ A ε θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q ě @ ξ P p , s . (26)In particular, it follows from (25) and (26) that there exists an ε P p , s such that for all ε P p , ε sxp W ´ A ε q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q ě ελ } θ ,ε ˆ ψ } L pr´ π,π s d q . (27) PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 15
3) Analysis of the localization error E . Since by Taylor expansion there exists a C ą | W p ξ q ´ W p ξ q| ď C | ξ | for | ξ | ď
2, one gets | E p ε q| “ ˇˇˇ xp W ´ W q θ ,ε ˆ ψ, θ ,ε ˆ ψ y L pr´ π,π s d q ˇˇˇ ď sup | ξ |ă ε | W p ξ q ´ W p ξ q| } θ ,ε ˆ ψ } L pr´ π,π s d q ď C ε } θ ,ε ˆ ψ } L pr´ π,π s d q @ ε P p , s . In particular, since } θ ,ε ˆ ψ } L pr´ π,π s d q ď } ˆ ψ } L pr´ π,π s d q “ } ψ } ℓ p ε Z d q , it followsthat E p ε q ě ´ C ε } ψ } ℓ p ε Z d q @ ε P p , s . (28)
4) Analysis of the localization error E . A straightforward computation (see also [43, Lemma 11.3]) gives the IMSlocalization formula A ε ˆ ψ ´ ÿ j “ θ j,ε A ε p θ j,ε ˆ ψ q “ ε ÿ j “ x ∇ θ j,ε , M ∇ θ j,ε y ˆ ψ p ξ q on R d . Thus there exists a constant C ą | E p ε q| ď ε ÿ j “ sup ξ P R d |x ∇ θ j,ε p ξ q , M ∇ θ j,ε p ξ qy| } ˆ ψ } L pr´ π,π s d q ď C ε ´ s } ˆ ψ } L pr´ π,π s d q @ ε P p , s . Recalling that s “ and the Parseval theorem (16) we conclude that E p ε q ě ´ C ε } ψ } ℓ p ε Z d q @ ε P p , s . (29) Final step.
Statement (i) in the theorem follows by putting together (20), (21), (22),(27), (28) and (29), chosing C “ C ` C and ε “ ε and observing that } θ ,ε ˆ ψ } L pr´ π,π s d ` } θ ,ε ˆ ψ } L pr´ π,π s d “ } ˆ ψ } L pr´ π,π s d “ } ψ } ℓ p ε Z d q . Statement (ii) follows similarly, but using (23) instead of (22) and chosingΨ ε “ θ ,ε Φ ε ˇˇˇ ε Z d . (cid:3) Laplace asymptotics on ε Z d . Given x P R d and δ ą B δ p x q “ t x P R d : | x ´ x | ă δ u theopen ball of radius δ around x and, for each ε ą
0, by B εδ p x q “ B δ p x qX ε Z d its intersection with ε Z d and by r B εδ p x qs c “ ε Z d z B εδ p x q the complementaryof B εδ p x q . Proposition 3.3.
Let q p x q “ x ¨ Qx , where Q is a symmetric, positivedefinite d ˆ d matrix and let x P R d and m P N . Then there exists a γ ą such that for every ε P p , s ε d ÿ x P ε Z d | x ´ x | m e ´ q p x ´ x q ε “ ε m ż R d | x ´ x | m e ´ q p x ´ x q dx ` O p e ´ γε q . (30) Moreover for every δ ą there exists γ p δ q ą such that for every ε P p , s ÿ x P B εδ p x q | x ´ x | m e ´ q p x ´ x q ε “ ÿ x P ε Z d | x ´ x | m e ´ q p x ´ x q ε ´ ` O p e ´ γ p δ q ε q ¯ . (31) Remark 3.4.
The Gaussian integrals appearing on the right hand sideof (30) can be computed explicitly. We shall use in the sequel the explicitvalue only for m “ , in which case (30) becomes ε d ÿ x P ε Z d e ´ q p x ´ x q ε “ c p π q d det Q ` O p e ´ γε q . We shall also use the following estimate for odd moments: ε d ÿ x P ε Z d | x ´ x | m e ´ q p x ´ x q ε “ O p ε m q for m “ , , . . . , (32) The latter follows from Proposition 3.3 and the Cauchy-Schwarz inequality ˇˇˇˇˇ ε d ÿ x P ε Z d | x ´ x | m e ´ q p x ´ x q ε ˇˇˇˇˇ ď ˜ ε d ÿ x P ε Z d | x ´ x | m e ´ q p x ´ x q ε ¸ ˜ ε d ÿ x P ε Z d e ´ q p x ´ x q ε ¸ . Proof of Proposition 3.3.
The function x ÞÑ u p x q : “ | x | m e ´ q p x ´ x q is in the Schwartz space S p R d q and its Fourier transform ˆ u p x q : “ ş R d u p y q e ´ π i x ¨ y dy satisfies the Poissonsummation formula (see e.g. [42, Corollary 2.6, p. 252]) ÿ x P Z d u p x q “ ÿ x P Z d ˆ u p x q . It follows that ε d ÿ x P ε Z d | x ´ x | m e ´ q p x ´ x q ε “ ε d ` m ÿ x P Z d |? ε p x ´ x ε q| m e q p? ε p x ´ x ε qq ““ ε d ` m ÿ x P Z d u p? ε p x ´ x ε qq “ ε m ÿ x P Z d e ´ πix ¨ x ε ˆ u p x ? ε q “ ε m ż R d u p x q dx ` R ε , with R ε : “ ε m ÿ x P Z d zt u e ´ πix ¨ x ε ˆ u p x ? ε q . Since ˆ u is a linear combination of derivatives of Gaussian functions, thereexist constants C, γ ą | ˆ u p x q| ď Ce ´ γ | x | @ x P R d . PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 17
It follows that for every ε P p , s| R ε | ď Cε m ÿ x P Z d zt u e ´ γ | x | ε “ Cε m e ´ γε ÿ x P Z d zt u e ´ γε p | x | ´ q ď C e ´ γε , with C : “ C ř x P Z d zt u e ´ γ p | x | ´ q which concludes the proof of (30).In order to prove (31), fix δ ą Q , there exists a constant C ą q p x q ą Cδ for every x P r B εδ p x qs c . Thus, for ε P p , s , ÿ x Pr B εδ p x qs c e ´ q p x ´ x q ε “ e ´ Cδ ε ÿ x Pr B εδ p x qs c e ´ q p x ´ x q´ Cδ ε ďď ε ´ d e ´ Cδ ε e Cδ ε d ÿ x Pr B εδ p x qs c e ´ q p x ´ x q ď ε ´ d e ´ Cδ ε K, (33)with K “ e Cδ `ş R d e ´ q p x ´ x q dx ` ˘ . To see the last inequality one canuse e.g. the Poisson summation formula for ε d ř x P ε Z d e ´ q p x ´ x q . From (33),chosing γ ą C ą ÿ x Pr B εδ p x qs c e ´ q p x ´ x q ε ď C e ´ γε . The estimate (31) for m “ m “
0. The caseof positive m can be proven in the same way. (cid:3) The following proposition concerns more general, not necessarily quadraticphase functions.
Proposition 3.5.
Let x P R d , δ ą , k P t , u and ϕ P C k p B δ p x qq s.t. ϕ p x q “ , Hess ϕ p x q ą and ϕ p x q ą for every x P B δ p x q . (34) Moreover let m P N . Then for ε P p , s it holds ε d ÿ x P B εδ p x q | x ´ x | m e ´ ϕ p x q ε “ ε m ż R d | x ´ x | m e ´ q p x ´ x q dx ˆ ` O p ε k ´ q ˙ , (35) where q p x q “ Hess ϕ p x q x ¨ x for all x P R d . Remark 3.6.
Under the stronger regularity assumption ϕ P C p B δ p x qq one can show that the error term in (35) admits a complete asymptoticexpansion in powers of ε , see [15, Appendix C] for details.Proof. We reduce the problem to the quadratic case of Proposition 3.3. For x P B δ p x q let for short r p x q “ ϕ p x q ´ q p x ´ x q and note that there exist α, δ ą q p x q : “ α | x | satisfies˜ ϕ p x q : “ q p x ´ x q ´ | r p x q| ě ˜ q p x ´ x q @ x P B δ p x q , (36)and also ϕ p x q ě ˜ q p x ´ x q for all x P B δ p x q . Indeed, the assumption ϕ P C p B δ p x qq implies the existence of a constant C ą | r p x q| ď C | x ´ x | for all x P B δ p x q . It follows that, denoting by λ ą ϕ p x q and taking e.g. δ “ λ C and α “ λ ,˜ ϕ p x q ě p λ ´ Cδ q| x ´ x | ě λ | x ´ x | @ x P B δ p x q . Note that, a fortiori, also ϕ p x q ě ˜ q p x q for every x P B δ p x q . Moreover, since ϕ p x q| x | is continuous and stricly positive on the compact set B δ p x qz B δ p x q wecan take e.g. α “ min t α , inf x P B δ p x qz B δ p x q ϕ p x q| x | u . It will be enough to prove (35) with the sum on the left hand side restrictedto B εδ p x q , since by Proposition 3.3 there exists a γ ą ε P p , s ÿ x P B εδ p x qz B εδ p x q | x ´ x | m e ´ ϕ p x q ε ď ÿ x Pr B εδ p x qs c | x ´ x | m e ´ ˜ q p x ´ x q ε “ O p e ´ γ { ε q . We shall consider the decomposition ε d ÿ x P B εδ p x q | x ´ x | m e ´ ϕ p x q ε “ I p ε q ` I p ε q ` I p ε q , (37)with, setting for short u ε p x q “ | x ´ x | m e ´ q p x q ε , I p ε q “ ε d ÿ x P B εδ p x q u ε p x q , I p ε q “ ε d ÿ x P B εδ p x q ε ´ r p x q u ε p x q and I p ε q “ ε d ÿ x P B εδ p x q ´ e ´ r p x q ε ´ ´ ε ´ r p x q ¯ u ε p x q . It follows from Proposition 3.3 that there exists a γ ą ε P p , s I p ε q “ ε m ż R d | x ´ x | m e ´ q p x ´ x q dx ` O p e ´ γε q . (38)Morever, using | r p x q| ď C | x ´ x | for all x P B δ p x q and (36) gives | I p ε q| ď ε d ´ ÿ x P B εδ p x q | r p x q| e | r p x q| ε u ε p x q ď C ε d ´ ÿ x P B εδ p x q | x ´ x | m ` e ´ ˜ q p x ´ x q ε “ O p ε m ` q , (39)with the last estimate being a consequence of Proposition 3.3. Finally, inorder to analyze the term I p ε q , we consider first the case k “
3. We thenhave by (32) | I p ε q| ď ε d ´ C ÿ x P B εδ p x q | x ´ x | m ` e ´ q p x ´ x q ε “ O p ε m ` q , which together with (37), (38) and (39) finishes the proof for k “
3. Forthe case k “ r p x q “ t p x q ` ρ p x q , where t : B δ p x q Ñ R is the cubic term in the Taylor expansion of ϕ around x , thus satisfying PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 19 t p x ` x q “ t p x ´ x q , and ρ : B δ p x q Ñ R satisfies | ρ p x q| ď C | x ´ x | for some C ą
0. We then have I p ε q “ ε d ´ ÿ x P B εδ p x q p t p x q ` ρ p x qq u ε p x q “ ε d ´ ÿ x P B εδ p x q ρ p x q u ε p x q , and therefore by Proposition 3.3 | I p ε q| ď ε d ´ C ÿ x P B εδ p x q | x ´ x | m ` e ´ q p x ´ x q ε “ O p ε m ` q , which finishes the proof in the case k “ (cid:3) Proof of Theorem 2.2
Recall the definition of V ε given in (6). To prove Theorem 2.2 we shall reduceto suitable localized problems and then exploit basic pointwise estimates on V ε as stated in the following two complementary lemmata. The first onegives 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 weakerassumptions, if instead of V ε one considers the corresponding continuousspace potential | ∇ f | ´ ε ∆ f appearing in (3). The discrete case followsfrom straightforward Taylor expansions and elementary estimates. We shallgive the details of the arguments at the end of this section for completeness. Lemma 4.1.
Assume f P C p R d q and that Hess f is bounded on R d . Let S Ă R d and a ą such that | ∇ f p x q| ą a for every x P S . Then there existconstants ε , C ą such that V ε p x q ě C @ x P S and @ ε P p , ε s . Lemma 4.2.
Assume f P C p R d q . Let z P R d such that ∇ f p z q “ , R ą and U p x q : “ xr Hess f p z qs p x ´ z q , p x ´ z qy . Then there exists a constant C ą such that for all x P B R p z q and ε ą | V ε p x q ´ U p x q ` ε f p z q| ď C ` | x ´ z | ` ε | x ´ z | ` ε ˘ . 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 bya constant, as claimed in Theorem 2.2 (i). Proposition 4.3 (Localization of the essential spectrum) . Under Assump-tion H1 there exist constants ε , C ą such that Spec ess p H ε q Ă r C,
8q @ ε P p , ε s . Remark 4.4.
The proof given below shows that the claim of Proposition 4.3still 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 p R d q instead of f P C p R d q . Proof.
Let χ : “ α K , where 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 p ε Z d q , isof finite rank (in particular compact) for every ε ą
0. It follows from Weyl’stheorem that for fixed ε ą ess ` H ε ˘ “ inf Spec ess ` H ε ` χ ˘ . (40)Moreover inf Spec ess ` H ε ` χ ˘ ě inf Spec ` H ε ` χ ˘ “ inf ψ P D om p V ε q ψ ‰ xp H ε ` χ q ψ, ψ y ℓ p ε Z d q x ψ, ψ y ℓ p ε Z d q ě inf ψ P D om p V ε q ψ ‰ xp V ε ` χ q ψ, ψ y ℓ p ε Z d q x ψ, ψ y ℓ p ε Z d q . The claim follows by chosing α and K large enough so that for some con-stants ε , C ą V ε p x q ` χ p x q ě C holds for every x P R d and ε P p , ε s . To see that this choice is possible recall the uniform bound V ε ě ´ d and note that by Assumption H1 (i) there exist a ą , R ą | ∇ f p x q| ą a for | x | ą R . It follows then by Lemma 4.1 that for suitable C, ε ą V ε p x q ě C for | x | ą R and ε P p , ε q . (cid:3) The next proposition provides the crucial estimate for the proof of statement(ii) in Theorem 2.2.
Proposition 4.5.
Assume H1 and denote by N P N the number of localminima of f . Then there exist constants ε , C ą and, for every ε ą ,functions Ψ ,ε , . . . , Ψ N ,ε P ℓ p ε Z d q such that for every ψ P D om p V ε q it holds x H ε ψ, ψ y ℓ p ε Z d q ě Cε } ψ } ℓ p ε Z d q ´ N ÿ k “ x ψ, Ψ k,ε y ℓ p ε Z d q @ ε P p , ε s . (41)Statement (ii) in Theorem 2.2 is then a simple consequence of the Max-Minprinciple (see e.g. [24, Theorem 11.7]): Corollary 4.6.
Assume H1 and denote by N P N the number of localminima of f . Then there exist constants ε , C ą such that | Spec disc p H ε q X r , Cε s| ď N @ ε P p , ε s . Proof of Corollary 4.6.
By Proposition 4.3 and Proposition 4.5 we can find ε , C ą ess p H ε q Ă r Cε ,
8q @ ε P p , ε s (42)and such that (41) holds. If for every ε P p , ε q it happens that | Spec disc p H ε q X r , Cε s| ď N , the claim is proven. Thus, we only have to check the case in which thereexists ε ˚ P p , ε q such that | Spec disc p H ε ˚ q X r , Cε ˚ s| ą N . (43)But this case is impossible. Indeed (43) implies that there exist at least N ` H ε ˚ in r , Cε ˚ s and thus in particular the N ` λ N ` p ε ˚ q (in increasing order and counting multiplicity) existsand satisfies λ N ` p ε ˚ q ď Cε ˚ . (44) PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 21
In particular λ N ` p ε ˚ q ď Cε and therefore, by (42), λ N ` p ε ˚ q is smallerthan the bottom of the essential spectrum. From this, the Max-Min principleand (41) it follows that λ N ` p ε ˚ q ě inf ψ x H ε ˚ ψ, ψ y ℓ p ε Z d q ě Cε ˚ , (45)where the infimum is taken over all normalized ψ P V K ε X D om p V ε q , with V ε being the linear span of the set t Ψ ,ε , . . . , Ψ N ,ε u Ă ℓ p ε Z d q appearingin (41). But (44) and (45) are in contradiction. (cid:3) Proof of Proposition 4.5.
We label by z , . . . , z N the critical points of f , theordering being chosen such that z , . . . , z N are the local minima. Then wetake a function χ P C p R d ; r , sq which equals 1 on t x : | x | ď u andvanishes on t x : | x | ě u . We shall consider a smooth quadratic partition ofunity by defining with s “ χ j,ε p x q : “ χ ` ε ´ s p x ´ z j q ˘ , χ ,ε p x q : “ ˜ ´ ÿ j χ j,ε p x q ¸ for j “ , . . . , N and ε P p , ε s , where ε P p , s is sufficiently small so that χ ,ε P C p R d q . We set moreover for x P R d and j “ , . . . , NU j p x q : “ xr Hess f p z j qs p x ´ z j q , p x ´ z j qy . Let ψ P D om p V ε q . It follows from ř Nj “ χ j,ε ” x H ε ψ, ψ y ℓ p ε Z d q “ N ÿ j “ x ` ´ ε ∆ ε ` U j ´ ε ∆ f p z j q ˘ χ j,ε ψ, χ j,ε ψ y ℓ p ε Z d q `x ` ´ ε ∆ ε ` V ε ˘ χ ,ε ψ, χ ,ε ψ y ℓ p ε Z d q ` E ` E , (46)with the localization errors given by E “ E p ε q : “ N ÿ j “ x ` V ε ´ U j ` ε ∆ f p z j q ˘ χ j,ε ψ, χ j,ε ψ y ℓ p ε Z d q , E “ E p ε q : “ ´ ε N ÿ j “ xp χ j,ε ∆ ε ´ ∆ ε χ j,ε q ψ, χ j,ε ψ y ℓ p ε Z d q . The four terms in the right hand side of (46) are now analyzed separately.
1) Analysis of the first term in the right hand side of (46) . We apply Proposition 3.2: let κ p z j q . . . , κ d p z j q be the eigenvalues of Hess f p z j q ,so that in particular ∆ f p z j q “ ř i κ i p z j q and κ p z j q . . . , κ d p z j q are theeigenvalues of r Hess f p z j qs . Case 1: j “ , . . . , N (i.e. z j is a local minimum of f ) In this case ř i p| κ i p z j q| ´ κ i p z j qq “ ε ą , j “ , . . . , N a function Φ j,ε P ℓ p ε Z d q and constants ε , C ą ε P p , ε s , j “ , . . . , N and ψ P C c p ε Z d qx ` ´ ε ∆ ε ` U j ´ ε ∆ f p z j q ˘ χ j,ε ψ, χ j,ε ψ y ℓ p ε Z d q ě C ε } χ j,ε ψ } ℓ p ε Z d q ´ x χ j,ε ψ, Φ j,ε y ℓ p ε Z d q . (47) Case 2: j “ N ` , . . . , N (i.e. z j is not a local minimum of f ) In this case ř i p| κ i p z j q| ´ κ i p z j qq ą ε , C ą ε Pp , ε s , j “ N ` , . . . , N and ψ P C c p ε Z d qx ` ´ ε ∆ ε ` U j ´ ε ∆ f p z j q ˘ χ j,ε ψ, χ j,ε ψ y ℓ p ε Z d q ě C ε } χ j,ε ψ } ℓ p ε Z d q . (48)
2) Analysis of the second term in the right hand side of (46) . According to Lemma (4.2) there exist constants r, C ą ε P p , ε s such that for every j “ , . . . , NV ε p x q ě C | x ´ z | ´ εC @ x P B r p z j q and @ ε P p , ε s . (49)Moreover, according to Lemma (4.1), possibly taking the constants ε , C ą V ε p x q ě C @ x P R d z N ď j “ B r p z j q and @ ε P p , ε s . (50)Since supp χ ,ε Ă t x P R d : | x ´ z j | ě ε for all j “ , . . . , N u , the lowerbounds (49), (50) imply (with possibly reducing further the constant ε ą x V ε χ ,ε ψ, χ ,ε ψ y ℓ p ε Z d q ě C ε } χ ,ε ψ } ℓ p ε Z d q @ ε P p , ε s . Using ´ ∆ ε ě x ` ´ ε ∆ ε ` V ε ˘ χ ,ε ψ, χ ,ε ψ y ℓ p ε Z d q ě C ε } χ ,ε ψ } ℓ p ε Z d q @ ε P p , ε s . (51)
3) Analysis of the localization error E . Let R j,ε p x q : “ V ε p x q´ U j p x q` ε ∆ f p z j q . By Lemma 4.2 there exist constants C , ε ą x : | x ´ z j |ď ε | R j,ε p x q| ď C ε @ ε P p , ε s and @ j “ , . . . , N. Thus, since supp χ j,ε Ă t x P R d : | x ´ z j | ď ε u for all j “ , . . . , N , | E p ε q| “ ˇˇˇˇˇ N ÿ j “ x R j,ε χ j,ε ψ, χ j,ε ψ y ℓ p ε Z d q ˇˇˇˇˇ ď C ε N ÿ j “ } χ j,ε ψ } ℓ p ε Z d q @ ε P p , ε s , and we conclude that E p ε q ě ´ C ε } ψ } ℓ p ε Z d q @ ε P p , ε s . (52) PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 23
4) Analysis of the localization error E . Using ř Nj “ χ j,ε ” E p ε q “ ´ ε x ˜ ∆ ε ´ N ÿ j “ χ j,ε ∆ ε χ j,ε ¸ ψ, ψ y ℓ p ε Z d q . Since there is a constant K ą x P R d | Hess χ j,ε p x q| ď Kε ´ s for every ε P p , ε s and j “ , . . . , N , it follows from Lemma 3.1 that thereexists a constant C ą | E p ε q| ď C ε ´ s } ψ } ℓ p ε Z d q “ C ε } ψ } ℓ p ε Z d q @ ε P p , ε s . In particular we shall use that E p ε q ě ´ C ε } ψ } ℓ p ε Z d q @ ε P p , ε s . (53) Final step.
Taking Ψ j,ε : “ χ j,ε Φ j,ε , ˜ ε : “ min t ε , ε , ε u and ˜ C : “ min t C , C u gives,according to (46), (47), (48), (51), (52), (53) the lower bound x H ε ψ, ψ y ℓ p ε Z d q ě ´ ˜ Cε ´ p C ` C q ε ¯ } ψ } ℓ p ε Z d q ´ N ÿ j “ x ψ, Ψ j,ε y ℓ p ε Z d q @ ε P p , ˜ ε s , which implies the desired estimate (41), by taking C “ ˜ C { ε P p , ˜ ε q . (cid:3) It remains to show part (iii) of Theorem 2.2 to complete the proof. In orderto do so, we can assume N ‰
0, since otherwise there is nothing to prove.By the Max-Min principle [24, Theorem 11.7 and Proposition 11.9] togetherwith the bound on the essential spectrum given by Proposition (4.3) it issufficient to show that for each ε ą N orthonormal functionsin the domain D om p V 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 exhibitsuch a family of orthonormal functions.Let t z , . . . , z N u be the set of local minima of f . We fix δ ą B δ p z k q X B δ p z j q is the empty set for k ‰ j and such that f ą f p z k q on B δ p z k qzt z k u . Moreover we fix for each k “ , . . . , N a cutoff function χ k P C p R d ; r , sq , satisfying χ ” B δ p z k q χ ” R d z B δ p z k q . Weconsider then for each ε ą k “ , . . . , N the functions ψ k,ε : R d Ñ R given by ψ k,ε p x q “ χ k p x q e ´ f p x q{p ε q } χ k e ´ f {p ε q } ℓ p ε Z d q . (54)Then for each ε ą ε Z d of the) functions ψ ,ε , . . . , ψ N ,ε are in the domain of H ε and orthonormal in ℓ p ε Z d q . Moreover the fol-lowing proposition shows that the quadratic form associated with H ε is exponentially small for each of these functions and thus concludes the proofof Theorem 2.2. Proposition 4.7.
Assume H1 and that the set t z , . . . , z N u of local minimaof f is not empty. Then there exist C, ε ą such that for each ε P p , ε s the functions ψ ,ε , . . . , ψ N ,ε defined in (54) satisfy the estimate x H ε ψ k,ε , ψ k,ε y ℓ p ε Z d q ď e ´ C { ε . Proof.
Fix k “ , . . . , N . Then, applying Proposition 3.5 with ϕ “ f ´ f p z k q , k “ m “
0, gives for a suitable constant K ą ε P p , s} χ k e ´ f {p ε q } ℓ p ε Z d q ě ε d e ´ f p z k q{ ε ÿ x P B εδ p z k q e ´p f p x q´ f p z k q{p ε q ě Kε d e ´ f p z k q{ ε . (55)Further, using (7) and the notation F ε p x, v q “ r f p x q ` f p x ` εv qs , x H ε p χ k e ´ f {p ε q q , χ k e ´ f {p ε q y ℓ p ε Z d q “ ε } e ´ F ε {p ε q ∇ ε χ k } ℓ p ε Z d ; R N q . (56)We take ε P p , s small enough such that for all ε P p , ε s it holds ∇ ε χ k p x, v q “ @ x P B δ { p z k q and @ v P N . Moreover we take γ ą ε P p , ε s and for all k “ , . . . , N it holds F ε p x, v q ´ f p z k q ě γ @ x P Ω ε : “ B δ p z k qz B δ { and @ v P N . It follows then from (56), the uniform bound ε | ∇ ε χ k | ď K ą ε d | Ω ε | ď ˜ K that x H ε p χ k e ´ f {p ε q q , χ k e ´ f {p ε q y ℓ p ε Z d q ď d ` ˜ Ke ´r f p z k q` γ s{ ε . (57)Putting together (55) and (57) gives the claim with e.g. C “ γ { ε P p , ε s sufficiently small. (cid:3) In the remainder of this section we provide the proofs of the basic estimateson 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 repre-sentation V ε p x q “ ÿ v P N sinh ∇ f p x q¨ v ` ε ÿ v P N e ´ ∇ f p x q¨ v R ε p x, v q , (58)where, thanks to the boundedness of Hess f , D R ą | R ε p x, v q| ď R @ x P R d , v P N and ε P p , s . In fact, one may write V ε p x q “ ÿ v P N „ e ´ ∇ f p x q¨ v ´ ` ε ÿ v P N e ´ ∇ f p x q¨ v ε „ e ´ f p x ` εv q´ f p x q´ ε ∇ f p x q¨ v ε ´ , PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 25 and, using cosh 2 t ´ “ t , ÿ v P N „ e ´ ∇ f p x q¨ v ´ “ ÿ v P N „ e ∇ f p x q¨ v ` e ´ ∇ f p x q¨ v ´ “ ÿ v P N ” cosh ∇ f p x q¨ v ´ ı “ ÿ v P N sinh ∇ f p x q¨ v . Moreover, for R ε p x, v q : “ ε „ e ´ f p x ` εv q´ f p x q´ ε ∇ f p x q¨ v ε ´ , using | e t ´ | ď | t | e | t | with t : “ ´ f p x ` εv q´ f p x q´ ε ∇ f p x q¨ v ε , and noting that, due to the boundedness of Hess f , there exists a constant A ą | t | ď ε sup x | Hess f p x q v ¨ v | ď ε A | v | , one gets for ε P p , s and every x P R d | R ε p x, v q| ď A | v | e A | v | ď max v P N A | v | e max v P N A | v | “ : R ą . (59)It follows from (58) and (59) that for ε P p , s and every x P R d V ε p x q ě ÿ v P N ” cosh ∇ f p x q¨ v ´ ı ´ εR ÿ v P N e ´ ∇ f p x q¨ v ““ ÿ v P N ” p ´ εR q p cosh ∇ f p x q¨ v ´ q ´ εR ı . Using that cosh t ´ ě t with t “ ∇ f p x q¨ v and ř v P N | ∇ f p x q¨ v | “ | ∇ f p x q| we get for ε P p , min t , R uq and every x P R d the lower bound V ε p x q ě „ p ´ εR q s | ∇ f p x q| ´ εR . In particular V ε p x q ě a ´ ε ˆ Ra ` R ˙ @ x P S and @ ε P p , min t , R uq . The claim follows by chosing an ε P p , min t , R , a Ra ` R uq and C “ a ´ ε ´ Ra ` R ¯ . (cid:3) Proof of Lemma 4.2.
This follows from a straightforward Taylor expansion.Indeed, fixing z P R d such that ∇ f p z q “ R ą
0, we have on B R p z q the uniform estimate ´ ε r f p¨ ` εv q ´ f s “ ´ ∇ f ¨ v ´ ε Hess f v ¨ v ` O p ε q . Using the inequality | e t ´ ´ t | ď t e | t | with t “ ε Hess f v ¨ v ` O p ε q thengives V ε “ ÿ v P N " e ´ ∇ f ¨ v ´ ´ e ´ ∇ f ¨ v ε Hess f v ¨ v ` O p ε q * “ ÿ v P N cosh r ∇ f ¨ v s ´ ´ cosh r ∇ f ¨ v s ε Hess f v ¨ v ` O p ε q ( . The expansion cosh x “ ` x ` O p x q and the equalities ř v | ∇ f ¨ v | “ | ∇ f | and ř v Hess f v ¨ v “ f give V ε “ | ∇ f | ` O p ÿ k |B k f | q ´ ε ∆ f ` O p ε | ∇ f | q ` O p ε q . Expanding all terms in x around z , which gives in particular | ∇ f p x q| “r Hess f p z qs p x ´ z q ¨ p x ´ z q ` O p| x ´ z | q and ∆ f p x q “ ∆ f p z q ` O p| x ´ z |q ,finishes the proof. (cid:3) Proof of Theorem 2.3
General strategy.
In order to compute the precise asymptotics of the smallest non-zero eigen-value λ p ε q of H ε we shall consider a suitable choice of an ε -dependent testfunction ψ ε . The latter will be referred to as quasimode and its precise con-struction will be given in Subsection 5.2. Since ψ ε will be chosen orthogonalto the ground state e ´ f {p ε q for every ε , the upper bound on λ p ε q given inTheorem 2.3 will follow immediately from the Max-Min principle, giving λ p ε q ď x H ε ψ ε , ψ ε y ℓ p ε Z d q } ψ ε } ℓ p ε Z d q , (60)and from the precise computation of the right hand side in the above formulaby using the Laplace asymptotics on ε Z d given in Subsection 3.3. The resultof these computations is the content of Proposition 5.2 and Proposition 5.3.The proof of the lower bound on λ p ε q given in Theorem 2.3 is more subtle.We shall derive it as a corollary of Theorem 2.2 and the following abstractestimate, which was used in [17] in a similar way. Proposition 5.1.
Let p T, D p T qq be a nonnegative selfadjoint operator on aHilbert space p X, x¨ , ¨yq . Moreover let τ ą and P “ r ,τ s p T q be the spectralprojector of T corresponding to the interval r , τ s and let λ “ sup pr , τ s X Spec p T qq . Then for every normalized u P D p T q with x T u, u y ‰ it holds λ ě x T u, u y p ´ R p u qq , where R p u q ě satisfies r R p u qs “ τ ´ x T u, T u yx T u, u y . PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 27
Proof.
We denote by } ¨ } the Hilbert space norm and fix a u P D p T q suchthat } u } “ x T u, u y ‰
0. Then we have the estimates λ ě } P u } λ “ x u, λP u y ě x u, T P u y “x T u, u y ˆ ´ x T u, u ´ P u yx T u, u y ˙ ě x T u, u y ˆ ´ } T u }} u ´ P u }x T u, u y ˙ . The claim follows now from the estimate } u ´ P u } ď τ ´ x T u, u y , which is a consequence of the spectral theorem. (cid:3) We shall apply Proposition 5.1 to the case T “ H ε , τ “ Cε , where C is theconstant appearing in Theorem 2.2 and u “ p} ψ ε } ℓ p ε Z d q q ´ ψ ε , where ψ ε isthe same quasimode used for the upper bound on λ p ε q . By Theorem 2.2(ii) we thus obtain a lower bound on λ p ε q . The fact that this lower boundcoincides with the lower bound given in Theorem 2.3 is a consequence ofthe precise computation of the right hand side of (60), which we alreadymentioned (see Prop. 5.2 and Prop. 5.3), and the estimate p Cε q ´ x H ε ψ ε , H ε ψ ε y ℓ p ε Z d q x H ε ψ ε , ψ ε y ℓ p ε Z d q “ O p ε q . The latter estimate will be a consequence of Proposition 5.3 and Proposi-tion 5.4, which is proven again by analyzing the Laplace asymptotics of asum over ε Z d .We shall assume throughout the rest of this section that the Assumption H2is satisfied.5.2. Definition of the quasimode ψ ε . Let s , . . . , s n be the relevant saddle points of f , i.e. the critical points ofindex 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 p x q aroundthe saddle point s k , which parametrizes the unstable direction of Hess f p s k q .More precisely, we chose one of the two normalized eigenvectors correspond-ing to the only negative eigenvalue µ p s k q of Hess f p s k q , denote it by τ k , andset ξ k p x q “ x x ´ s k , τ k y @ k “ , . . . , n. (61)Recalling our notation S f p h q “ f ´ pp´8 , h qq for the open sublevel set of f corresponding to the height h P R , we consider for ρ ą k “ , . . . , n the closed set R k “ ! x P S f p h ˚ ` ρ q : | ξ k p x q| ď ρ ) , and the open set B “ S f p h ˚ ` ρ qz p Ť k R k q .Henceforth the parameter ρ ą R k and B isfixed sufficiently small such that the following properties hold:- the set B has exactly two connected components B p q and B p q , con-taining respectively m and m .- R k is disjoint from R k for k ‰ k .- For each k “ , . . . , n the function ϕ k : “ f ` | µ p s k q| ξ k satisfies ϕ k p x q ą f p s k q for every x P R k zt s k u . Note that Hess ϕ k p s k q “ | Hess f p s k q| . In other terms the quadratic approx-imation of ϕ k around s k is obtained from that of f by flipping the sign ofthe only negative eigenvalue of Hess f p s k q .Let ε P p , s . The quasimode ψ ε for the spectral gap is defined as follows.We define first on the sublevel set S f p h ˚ ` ρ q κ ε p x q “ $’’&’’% ` x P B p q , ´ x P B p q ,C k,ε ş ξ k p x q χ p η q e ´ | µ p sk q| η ε dη for x P Ť k R k . The constant C k,ε appearing above is defined as C k,ε : “ „ ż χ p η q e ´ | µ p sk q| η ε dη ´ , and χ P C p R ; r , sq satisfies χ ” r´ ρ , ρ s , χ p η q “ | η | ě ρ and χ p η q “ χ p´ η q . Note that D γ ą C k,ε “ c | µ p s k q| πε ´ ` O p e ´ γε q ¯ . (62)Note also that for each k “ , . . . n the sign of the vector τ k defining ξ k (see (61)) can be chosen such that κ ε is C on S f p h ˚ ` ρ q , which we shallassume in the sequel. In order to extend κ to a smooth function definedon the whole R d we introduce another cutoff function θ P C p R d ; r , sq bysetting for x P R d θ p x q “ x P S f p h ˚ ` ρ q x P R d z S f p h ˚ ` ρ q . Finally we define the quasimode ψ ε by setting for x P R d ψ ε p x q “ ˆ θ p x q κ ε p x q ´ x θκ ε ,e ´ f { ε y ℓ p ε Z d q } e ´ f {p ε q } ℓ p ε Z d q ˙ e ´ f {p ε q . (63)Note that ψ ε P C p R d q with compact support. In particular its restictionto ε Z d , which we still denote by ψ ε , is in C c p ε Z d q Ă D om p V ε q . Moreover,it follows from its very definition that ψ ε is orthogonal to the ground state e ´ f {p ε q with respect to the scalar product x¨ , ¨y ℓ p ε Z d q .5.3. Quasimode estimates.
We now state the crucial estimates concerning the quasimode ψ ε . The proofsfollow from straightforward computations exploiting the results of Subsec-tion 3.3 on the Laplace asymptotics for sums over ε Z d . We shall give thedetails in Subsection 5.4. Proposition 5.2.
Assume H2 and let ε P p , s . The function ψ ε definedin (63) satisfies } ψ ε } ℓ p ε Z d q “ p πε q d J e ´ h ˚ { ε ` ` O p? ε q ˘ , PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 29 where h ˚ “ min t f p m q , f p m qu is the minimum of f and J “ $&%´ p det Hess f p m qq ` p det Hess f p m qq ¯ ´ if f p m q “ f p m q , p det Hess f p m qq ´ if f p m q ă f p m q . Proposition 5.3.
Assume H2 and let ε P p , s . The function ψ ε definedin (63) satisfies x H ε ψ ε , ψ ε y ℓ p ε Z d q “ ε n ÿ k “ | µ p s k q| π p πε q d | det Hess f p s k q| e ´ h ˚ { ε ` ` O p? ε q ˘ , where µ p s k q is the only negative eigenvalue of Hess f p s k q and h ˚ is definedin (9) . Proposition 5.4.
Assume H2 and let ε P p , s . The function ψ ε definedin (63) satisfies } H ε ψ ε } ℓ p ε Z d q “ O p ε q e ´ h ˚ { ε , where h ˚ is defined in (9) . Remark 5.5.
With the stronger assumption f P C p R d q the O p? ε q errorterms appearing in Proposition 5.2 and Proposition 5.2 can be shown tobe actually O p ε q . Indeed it is enough to apply Proposition 3.5 with k “ instead of k “ each time it is used in the proofs given below. Proofs of the quasimode estimates.
Proof of Proposition 5.2.
Let ε P p , s . We first consider the case f p m q ă f p m q . Then there exist α, δ ą f ě f p m q ` α on r B δ p m qs c and θκ ε ” ´ B δ p m q . It follows that, denoting for short by Ω δε the boundedset r B εδ p m qs c X supp p θ q , it holds } e ´ f {p ε q } ℓ p ε Z d q “ ε d ÿ x P B εδ p m q e ´ f p x q{ ε ` e ´r f p m q` α s{ ε ε d ÿ x P Ω δε e ´r f p x q´ α s{ ε “ ε d ÿ x P B εδ p m q e ´ f p x q{ ε ´ ` O p e ´r f p m q` α s{ ε q ¯ . Proposition 3.5 gives then } e ´ f {p ε q } ℓ p ε Z d q “ p πε q d p det Hess f p m qq ´ e ´ f p m q ` ` O p? ε q ˘ . (64)The same arguments and the estimate | θκ ε | ď x θκ ε , e ´ f { ε y ℓ p ε Z d q “ ´p πε q d p det Hess f p m qq ´ e ´ f p m q ` ` O p? ε q ˘ . (65)Taking the quotient between (64) and (65) it follows then from the definitionof ψ ε that ψ ε p x q “ ˆ θ p x q κ ε p x q ` ` O p? ε q ˙ e ´ f {p ε q . The norm ψ ε } ℓ p ε Z d q can be now computed by splitting again the sum intwo sums, respectively over B εδ p m q and Ω εδ . The conclusion in the case f p m q ă f p m q follows by again using Proposition 3.5 for the first sum andarguing as above for the second sum.We now consider the case f p m q “ f p m q . It follows from the definition of ψ ε that } ψ ε } ℓ p ε Z d q “ } θκe ´ f {p ε q } ℓ p ε Z d q ´ x θκ, e ´ f { ε y ℓ p ε Z d q } e ´ f {p ε q } ℓ p ε Z d q . (66)Let α, δ ą f ě f p m q ` α on r B δ p m q Y B δ p m qs c and θκ ε ” ´ B δ p m q , θκ ε ” B δ p m q . With arguments as above one gets } e ´ f {p ε q } ℓ p ε Z d q “ } θκe ´ f {p ε q } ℓ p ε Z d q ` ` O p? ε q ˘ ““ p πε q d ” p det Hess f p m qq ´ ` p det Hess f p m qq ´ ı e ´ f p m q{ ε ` ` O p? ε q ˘ , x θκ, e ´ f { ε y ℓ p ε Z d q ““ p πε q d ” p det Hess f p m qq ´ ´ p det Hess f p m qq ´ ı e ´ f p m q{ ε ` ` O p? ε q ˘ . Putting these expressions into (66), the desired result (5.2) follows aftersome algebraic manipulations. (cid:3)
Proof of Proposition 5.3.
Let ε P p , s . Using (7) and the notation F ε p x, v q “ r f p x q ` f p x ` εv qs gives x H ε ψ ε , ψ ε y ℓ p ε Z d q “ ε } e ´ F ε {p ε q ∇ ε p θκ ε q} ℓ p ε Z d ; R N q . Since the function θ has support in S f p h ˚ ` ρ q , we can restrict (for ε suffi-ciently small) the sum running over ε Z d to the bounded set ε Z d X S f p h ˚ ` ρ q .Note that S f p h ˚ ` ρ q is the union of the disjoint sets B and Ť k p R k zr S f p h ˚ ` ρ qs c q . We write in the sequel for short R k,ε : “ ε Z d X p R k zr S f p h ˚ ` ρ qs c q and B ε : “ ε Z d X B and discuss below separately the sum over Y nk “ R k,ε ,which will give the main contribution, and the sum over B ε , which will givea negligible contribution.Below we shall use the Taylor expansion e ´ F ε p x,v q{ ε “ e ´ f p x q{ ε e ´ ∇ f p x q¨ v { p ` O p ε qq . (67)
1) Analysis on Y nk “ R k,ε . In order to get rid of θ we take δ ą k itholds B εδ p s k q Ă R k,ε X S f p h ˚ ` ρ q Ă R k,ε . Since θ and κ ε are uniformlybounded in ε and f ě h ˚ ` ρ on R k,ε z B εδ p s k q , we get using (67) that, for The subtraction of r S f p h ˚ ` ρ qs c is necessary to have disjoint sets, but is not reallyrelevant, since it concerns only boundary terms which do not matter in the computationsgiven below.PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 31 ε ą ε P p , s ), it holds ε d ÿ x P R k,ε ÿ v P N r θκ ε p x ` εv q ´ θκ ε p x qs e ´ F ε p x,v q{ ε “ ε d ÿ x P B εδ p s k q ÿ v P N r κ ε p x ` εv q ´ κ ε p x qs e ´ F ε p x,v q{ ε ` O p e ´p h ˚ ` ρ q q , (68)where we have used also that for ε sufficiently small θ p x q “ θ p x ` εv q “ x P B εδ p s k q .We discuss now in detail the behavior of x ÞÑ κ ε p x ` εv q ´ κ ε p x q near s k .For k “ , . . . , n and x P R k , v P N and ε P p , s consider the function G “ G k,x,v,ε : r , s Ñ R defined by G p δ q “ C ´ k,ε r κ ε p x ` δv q ´ κ ε p x qs “ ż ξ k p x ` δv q ξ k p x q χ p η q e ´| µ p s k q| η {p ε q dη. (69)Note that G p q “ G p q “ e ´| µ p s k q| ξ k p x q{p ε q χ p ξ k p x qq τ k ¨ v , G p q “ e ´| µ p s k q| ξ k p x q{p ε q | τ k ¨ v | „ χ p ξ k p x qq ´ | µ p s k q| ξ k p x q ε χ p ξ k p x qq , and for every δ P r , s ε G p δ q “ εe ´| µ p s k q| ξ k p x ` δv q{p ε q p τ k ¨ v q “ | µ p s k q| ξ k p x ` δv q χ p ξ k p x ` δv q ` εR ‰ , where R is not depending on ε and bounded in k, x, v . By Taylor expansionit follows that G p ε q “ εe ´| µ p s k q| ξ k p x q{p ε q χ p ξ k p x qq ˆ (70) „ τ k ¨ v ´ | µ p s k q| ξ k p x q| τ k ¨ v | ` O p| x ´ s k | q p ` O p ε qq . It follows from (69), (70), (67), (62) and the two identities ÿ v P N | τ k ¨ v | e ´ ∇ f p x q¨ v { “ d ÿ j “ p e j ¨ τ k q cosh B j f p x q , (71) ÿ v P N p τ k ¨ v q e ´ ∇ f p x q¨ v { “ ´ ÿ j p e j ¨ τ k q sinh B j f p x q that for k “ , . . . , n and x P R k and ε ą ÿ v P N r κ ε p x ` εv q ´ κ ε p x qs e ´ F ε p x,v q{ ε “ (72) εe ´ f p s k q{ ε | µ p s k q| π e ´ ϕ k p x q{ ε α k p x q ` ` O p ε q ` O ` | x ´ s k | ˘˘ , where for shortness we have set ϕ k p x q “ f p x q ´ f p s k q ` | µ p s k q| ξ k p x q and α k p x q “ χ p ξ k p x qq d ÿ j “ " p e j ¨ τ k q cosh B j f p x q ` | µ p s k q| ξ k p x qp e j ¨ τ k q sinh B j f p x q * “ ` O p| x ´ s k | q . Putting together (68), (72), using Proposition 3.5, summing over k and thefact that f p s k q “ h ˚ for every k finally gives ε d ÿ x PY k R k,ε ÿ v P N r κ ε p x ` εv q ´ κ ε p x qs e ´ F ε p x,v q{ ε “ ε n ÿ k “ | µ p s k q| π p πε q d | det Hess f p s k q| e ´ h ˚ ε ` ` O p? ε q ˘ .
2) Analysis on B ε . As in Step 1) we get rid of θ by considering the set B ε “ B ε X S f p h ˚ ` ρ q Ă B .Arguing as before and now using that κ ε p x q “ κ ε p x ` εv q for every x P B ε , v P N and ε sufficiently small, gives then ε d ÿ x P B ε ÿ v P N r θκ ε p x ` εv q ´ θκ ε p x qs e ´ F ε p x,v q{ ε “ ε d ÿ x P B ε ÿ v P N r κ ε p x ` εv q ´ κ ε p x qs e ´ F ε p x,v q{ ε ` O p e ´p h ˚ ` ρ q q“ O p e ´p h ˚ ` ρ q q . (cid:3) Proof of Proposition 5.4.
The isomporphism (12) gives the identity } H ε ψ ε } ℓ p ε Z d q “ } ε Φ ε “ L ε Φ ´ ε r ψ s ‰ } ℓ p ρ ε q “ ε d ÿ x P ε Z d ˜ ÿ v P N e ´ ∇ ε f p x,v q ε ∇ ε p θκ ε qp x, v q ¸ e ´ f p x q ε . Since the function θ has support in S f p h ˚ ` ρ q , we can restrict (for ε sufficiently small) the sum over ε Z d to the bounded set ε Z d X S f p h ˚ ` ρ q .As in the proof of Proposition 5.3 we shall split the latter into the disjointsets Y k R k,ε ,with R k,ε : “ ε Z d X p R k zr S f p h ˚ ` ρ qs c q , and B ε : “ ε Z 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) ofProposition 5.3 and using instead of (67) that by Taylor expansion e ´ ∇ ε f p x,v q “ e ´ ∇ f p x q¨ v { p ` O p ε qq . (73) Analysis on Y nk “ R k,ε . PECTRAL ANALYSIS OF DISCRETE METASTABLE DIFFUSIONS 33
As in the proof of Proposition 5.3 we first get rid of θ by taking a δ ą k it holds B εδ p s k q Ă R k,ε X S f p h ˚ ` ρ q Ă R k,ε .Since θ and κ ε are uniformly bounded in ε and f ě h ˚ ` ρ on R k,ε z B εδ p s k q ,we get using (73) that, for ε ą ε Pp , s ), it holds ε d ÿ x P R k,ε ˜ ÿ v P N e ´ ∇ ε f p x,v q ε ∇ ε p θκ ε qp x, v q ¸ e ´ f p x q{ ε “ (74) ε d ÿ x P B εδ p s k q ˜ ÿ v P N e ´ ∇ ε f p x,v q ε ∇ ε κ ε p x, v q ¸ e ´ f p x q{ ε ` O p e ´p h ˚ ` ρ q q . A computation already used in the proof of Proposition 5.3 (see (70)) yields e | µ p s k q| ξ k p x q{p ε q C ´ k,ε ε ∇ ε κ ε p x, v q “ εχ p ξ k p x qq „ τ k ¨ v ´ | µ p s k q| ξ k p x q| τ k ¨ v | ` O p| x ´ s k | q p ` O p ε qq . Hence, using (62), (73), the identity (71) and the identity ÿ v P N τ k ¨ ve ´ ∇ f p x q¨ v { “ ´ ÿ j e j ¨ τ k sinh B j f p x q , one obtains e | µ p s k q| ξ k p x q{p ε q ÿ v P N e ´ ∇ ε f p x,v q ε ∇ ε κ ε p x, v q “ ? εα k p x q p ` O p ε qq , (75)with α k p x q “ ´ b | µ p s k q| π χ p ξ k p x qq ˆ d ÿ j “ ” e j ¨ τ k sinh B j f p x q ` | µ p s k q| ξ k p x qp e j ¨ τ k q cosh B j f p x q ` O p| x ´ s k | q ı . (76)Observing that d ÿ j “ e j ¨ τ k sinh B j f p x q “ x Hess f p s k q τ k , x ´ s k y ` O p| x ´ s k | q “´| µ p s k q| ξ k p x q ` O p| x ´ s k | q , and that d ÿ j “ p e j ¨ τ k q cosh B j f p x q “ ` O p| x ´ s k | q shows that the first order terms in (76) cancel out and thus α k p x q “ O p| x ´ s k | q . It follows then from (74), (75) that there exists a constant C ą that for every ε P p , s and every k “ , . . . , n ε d ÿ x P R k,ε ˜ ÿ v P N e ´ ∇ ε f p x,v q ε ∇ ε p θκ ε qp x, v q ¸ e ´ f p x q{ ε ď Cε d ` e ´ f p s k q{ ε ÿ x P B εδ p s k q | x ´ s k | e ´ ϕ k p x q{ ε “ O p ε q e ´ h ˚ { ε , with ϕ k p x q “ f p x q`| µ p s k q| ξ k p x q´ f p s k q and with the last estimate followingfrom Proposition 3.5 by taking m “ (cid:3) Acknowledgements:
The author gratefully acknowledges the financialsupport of HIM Bonn in the framework of the 2019 Junior Trimester Pro-grams “Kinetic Theory” and “Randomness, PDEs and Nonlinear Fluctua-tions”.
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Giacomo Di Ges`u, Univerist`a di Pisa, Largo Bruno Pontecorvo 5, 56127Pisa, Italy.
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