Sharp asymptotics for metastability in the random field Curie-Weiss model

In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.


INTRODUCTION AND MAIN RESULTS
The simplest example of disordered mean field models is the random field Curie-Weiss model. Here the state space is S N = {−1, 1} N , where N is the number of particles of the system. Its Hamiltonian is where Λ ≡ {1, . . . , N} and h i , i ∈ Λ, are i.i.d. random variables on some probability space (Ω, F , P h ). For sake of convenience, we will assume throughout this paper that the common distribution of h has bounded support.
The dynamics of this model has been studied before: dai Pra and den Hollander studied the short-time dynamics using large deviation results and obtained the analog of the McKeane-Vlasov equations [16]. Mathieu and Picco [15] and Fontes, Mathieu, and Picco [12], considered convergence to equilibrium in a particularly simple case where the random field takes only the two values ±ε. Finally, Bovier et al. [6] analyzed this model in the case when h takes finitely many values, as an example of the use of the potential theoretic approach to metastability. In this article we extend this analysis to the case of random fields with continuous distributions, while at the same time improving the results by giving sharp estimates of transition times between metastable states.
The present paper should be seen, beyond the interest presented by the model as such, as a first case study in the attempt to derive precise asymptotics of metastable characteristics in kinetic Ising models in situations where neither the temperature tends to zero nor an exact reduction to low-dimensional models is possible. While the RFCW model is certainly one of the simplest examples of this class, we feel that the general methodology developed here will be useful in a much wider class of systems.
1.1. Gibbs measure and order parameter. The static picture. The equilibrium statistical mechanics of the RFCW model was analyzed in detail in [1] and [13]. We give a very brief review of some key features that will be useful later. As usual, we define the Gibbs measure of the model as the random probability measure where the partition function is defined as We define the total magnetization as The magnetization will be the order parameter of the model, and we define its distribution under the Gibbs measures as the induced measure, where For simplicity we will in the sequel identify functions defined on the discrete set Γ N with functions defined on [−1, 1] by setting f (m) ≡ f ([2Nm]/2N). Then, for m ∈ (−1, 1), Z 1 N (m) can be expressed, using sharp large deviation estimates [10], as (1. 10) Here I N [ω](y) is the Legendre-Fenchel transform of the log-moment generating function Above we have indicated the random nature of all functions that appear by making their dependence on the random parameter ω explicit. To simplify notation, in the sequel this dependence will mostly be dropped. We are interested in the behavior of this function near critical points of F β,N . An important consequence of Equations (1.6) through (1.11) is that if m * is a critical point of F β,N , then for |v| ≤ N −1/2+δ , .
(1. 17) We see that, by the law of large numbers, the set of critical points converges, P halmost surely, to the set of solutions of the equation (The cases where βE h 1 − tanh 2 (β(m * + h)) = 1 correspond to second order phase transitions and will not be considered here). From this discussion we get a very precise picture of the distribution of the order parameter.
1.2. Glauber dynamics. We will consider for definiteness discrete time Glauber dynamics with Metropolis transition probabilities if σ and σ ′ differ on a single coordinate, , (1.25) and p N (σ, σ ′ ) = 0 in all other cases. We will denote the Markov chain corresponding to these transition probabilities σ(t) and write P ν [ω] ≡ P ν , for the law of this chain with initial distribution ν, and we will set P σ ≡ P δσ . As is well known, this chain is ergodic and reversible with respect to the Gibbs measure µ β,N [ω], for each ω. Note that we might also study chains with different transition probabilities that are reversible with respect to the same measures. Details of our results will depend on this choice. The transition matrix associated with these transition probabilities will be called P N , and we will denote by L N ≡ P N − ½ the (discrete) generator of the chain.
Our main result will be sharp estimates for mean hitting times between minima of the function F β,N (m) defined in (1.10).
More precisely, for any subset A ⊂ S N , we define the stopping time (1.26) We also need to define, for any two subsets A, B ⊂ S N , the probability measure on A given by . (1.27) We will be mainly concerned with sets of configurations with given magnetization. For any I ∈ Γ N , we thus introduce the notation S[I] ≡ {σ ∈ S N : m N (σ) ∈ I} and state the following: whereγ 1 is the unique negative solution of the equation β(1+tanh(β(z * +h))) − 2γ   = 1. (1.29) Note that we have the explicit representation for the random quantity The proof of this result on mean transition times relies on the following result on capacities (for a definition see Eq. (2.5) in Section 2 below). (1.31) The proof of Theorem 1.3 is the core of the present paper. As usual, the proof of an upper bound of the form (1.31) will be relatively easy. The main difficulty is to prove a corresponding lower bound. The main contribution of this paper is to provide a method to prove such a lower bound in a situation where the entropy of paths cannot be neglected.
Before discussing the methods of proof of these results, it will be interesting to compare this theorem with the prediction of the simplest uncontrolled approximation. The naive approximation. A widespread heuristic picture for metastable behavior of systems like the RFCW model is based on replacing the full Markov chain on S N by an effective Markov chain on the order parameter, i.e. by a nearest neighbor random walk on Γ N with transition probabilities that are reversible with respect to the induced measure, Q β,N . The ensuing model can be solved exactly. In the absence of a random magnetic field, this replacement is justified since the image of σ(t), m(t) ≡ m N (σ(t)), is a Markov chain reversible w.r.t. Q β,N ; unfortunately, this fact relies on the perfect permutation symmetry of the Hamiltonian of the Curie-Weiss model and fails to hold in the presence of random field.
A natural choice for the transition rates of the heuristic dynamics is (1.32) which are different from zero only if m ′ = m ± 2/N or if m = m ′ . The ensuing Markov process is a one-dimensional nearest neighbor random walk for which most quantities of interest can be computed quite explicitly by elementary means (see e.g. [17,3]). In particular, it is easy to show that for this dynamics, where a(z * ) is defined in (1.19). The prediction of the naive approximation is slightly different from the exact answer, albeit only by a wrong prefactor. One may of course consider this as a striking confirmation of the quality of the naive approximation; from a different angle, this shows that a true understanding of the details of the dynamics is only reached when the prefactors of the exponential rates are known (see [14] for a discussion of this point).
The picture above is in some sense generic for a much wider class of metastable systems: on a heuristic level, one wants to think of the dynamics on metastable time scales to be well described by a diffusion in a double (or multi) well potential. While this cannot be made rigorous, it should be possible to find a family of mesoscopic variables with corresponding (discrete) diffusion dynamics that asymptotically reproduce the metastable behavior of the true dynamics. The main message of this paper is that such a picture can be made rigorous within the potential theoretic approach. Acknowledgments. The authors thank Alexandre Gaudillière, Frank den Hollander, and Cristian Spitoni for useful discussions on metastability.

SOME BASIC CONCEPTS FROM POTENTIAL THEORY
Our approach to the analysis of the dynamics introduced above will be based on the ideas developed in [6,7,8] to analyze metastability through a systematic use of classical potential theory. Let us recall the basic notions we will need.
For two disjoint sets A, B ⊂ S N , the equilibrium potential, h A,B , is the harmonic function, i.e. the solution of the equation The equilibrium measure is the function which clearly is non-vanishing only on A and B. An important formula is the discrete analog of the first Green's identity: Let D ⊂ S N and D c ≡ S N \ D. Then, for any function f , we have 1 2 In particular, for f = h A,B , we get that where the right-hand side is called the capacity of the capacitor A, B. The functional appearing on the left-hand sides of these relations is called the Dirichlet form or energy, and denoted As a consequence of the maximum principle, the function h A,B is the unique minimizer of Φ N with boundary conditions (2.2), which implies the Dirichlet principle: where H A,B denotes the space of functions satisfying (2.2). Equilibrium potential and equilibrium measure have an immediate probabilistic interpretation, namely An important observation is that equilibrium potentials and equilibrium measures also determine the Green's function. In fact (see e.g. [7,4]), In the case then A is a single point, this relation can be solved for the Green's function to give This equation is perfect if the cardinality of the state space does not grow too fast. In our case, however, it is of limited use, since both numerator and denominator tend to be very close to zero for the wrong reason. However, (2.9) remains useful. In particular, it gives the following representation for mean hitting times 11) or, using definition (1.27) From these equations we see that our main task will be to obtain precise estimates on capacities and some reasonably accurate estimates on equilibrium potentials. In previous applications [6,7,8,9,5], three main ideas were used to obtain such estimates: (i) Upper bounds on capacities can be obtained using the Dirichlet variational principle with judiciously chosen test functions. (ii) Lower bounds were usually obtained using the monotonicity of capacities in the transition probabilities (Raighley's principle). In most applications, reduction of the network to a set of parallel 1-dimensional chains was sufficient to get good bounds. (iii) The simple renewal estimate h A,B (x) ≤ cap(x,A) cap(x,B) was used to bound the equilibrium potential through capacities again.
These methods were sufficient in previous applications essentially because entropy were not an issue there. In the models at hand, entropy is important, and due to the absence of any symmetry, we cannot use the trick to deal with entropy by a mapping of the model to a low-dimensional one, as can be done in the standard Curie-Weiss model and in the RFCW model when the magnetic field takes only finitely many values [15,6].
Thus we will need to improve on these ideas. In particular, we will need a new approach to lower bounds for capacities. This will be done by exploiting a dual variational representation of capacities in terms of flows, due to Berman and Konsowa [2]. Indeed, one of the main messages of this paper is to illustrate the power of this variational principle. Random path representation and lower bounds on capacities. It will be convenient to think of the quantities µ(σ)p N (σ, σ ′ ) as conductances, c(σ, σ ′ ), associated to the edges e = (σ, σ ′ ) of the graph of allowed transitions of our dynamics. This interpretation is justified since, due to reversibility, c(σ, σ ′ ) = c(σ ′ , σ) is symmetric.
For purposes of the exposition, it will be useful to abstract from the specific model and to consider a general finite connected graph, (S, E) such that whenever e = (a, b) ∈ E, then also −e ≡ (b, a) ∈ E. Let this graph be endowed with a symmetric function, c : E → R + , called conductance.
Given two disjoint subsets A, B ⊂ S define the capacity, (2.13) Definition 2.1. Given two disjoint sets, A, B ⊂ S, a non-negative, cycle free unit flow, f , from A to B is a function f : E → R + ∪ {0}, such that the following conditions are verified: (iv) any path, γ, from A to B such that f (e) > 0 for all e ∈ γ, is self-avoiding.
We will denote the space of non-negative, cycle free unit flows from A to B by U A,B .
An important example of a unit flow can be constructed from the equilibrium potential, h * , i.e. the unique minimizer of (2.13). Since h * satisfies, for any a ∈ S \ (A ∪ B), b c(a, b)(h * (b) − h * (a)) = 0, (2.16) one verifies easily that the function, f * , defined by is a non-negative unit flow from A to B. We will call f * the harmonic flow.
The key observation is that any f ∈ U A,B gives rise to a lower bound on the capacity cap(A, B), and that this bound becomes sharp for the harmonic flow. To see this we construct from f a stopped Markov chain X = (X 0 , . . . , X τ ) as follows: We define the initial distribution of our chain as P f (a) = F (a), for a ∈ A, and zero otherwise. The transition probabilities are given by for a ∈ B, and the chain is stopped on arrival in B. Notice that by our choice of the initial distribution and in view of (2.18) X will never visit sites a ∈ S \ B with F (a) = 0. Thus, given a trajectory X = (a 0 , a 1 , . . . , a r ) with a 0 ∈ A, a r ∈ B and a ℓ ∈ S \ (A ∪ B) for ℓ = 0, . . . , r − 1, , (2.19) where e ℓ = (a ℓ , a ℓ+1 ) and we use the convention 0/0 = 0. Note that, with the above definitions, the probability that X passes through an edge e is (2.20) Consequently, we have a partition of unity, We are ready now to derive our f -induced lower bound: For every function h with h| A = 0 and h| B = 1, As a result, interchanging the minimum and the sum, (2.24)

COARSE GRAINING AND THE MESOSCOPIC APPROXIMATION
The problem of entropy forces us to investigate the model on a coarse grained scale. When the random fields take only finitely many values, this can be done by an exact mapping to a low-dimensional chain. Here this is not the case, but we can will construct a sequence of approximate mappings that in the limit allow to extract the exact result.
3.1. Coarse graining. Let I denote the support of the distribution of the random fields. Let I ℓ , with ℓ ∈ {1, . . . , n}, be a partition of I such that, for some C < ∞ and for all ℓ, |I ℓ | ≤ C/n ≡ ε.
Each realization of the random field {h i [ω]} i∈N induces a random partition of the set Λ ≡ {1, . . . , N} into subsets We may introduce n order parameters We denote by m [ω] the n-dimensional vector (m 1 [ω], . . . , m n [ω]). In the sequel we will use the convention that bold symbols denote n-dimensional vectors and their components, while the sum of the components is denoted by the corresponding plain symbol, e.g. m ≡ n ℓ=1 m ℓ . m takes values in the set where We will denote by e ℓ , ℓ = 1, . . . , n, the lattice vectors of the set Γ n N , i.e. the vectors of length 2/N parallel to unit vectors.
Note that the random variables ρ N,k concentrate exponentially (in N) around their mean values E h ρ N,k = P h [h i ∈ I k ] ≡ p k .
Notational warning: To simplify statements in the remainder of the paper, we will henceforth assume that all statements involving random variables on (Ω, F , P h ) hold true with P h -probability one, for all but finitely many values of N.
We may write the Hamiltonian in the form where E : R n → R is the function Note that if h i =h ℓ for all i ∈ Λ ℓ , which is the case when h takes only finitely many values and the partition I ℓ is chosen suitably, then the Glauber dynamics under the family of functions m ℓ is again Markovian. This fact was exploited in [15,6]. Here we will consider the case where this is not the case. However, the idea behind our approach is to exploit that by choosing n large we can get to a situation that is rather close to that one. Let us define the equilibrium distribution of the variables m[σ] is the normalizing partition function. Note that with some abuse of notation, we will use the same symbols Q β,N , F β,N as in Section 1 for functions defined on the n-dimensional variables x. Since we distinguish the vectors from the scalars by use of bold type, there should be no confusion possible. Similarly, for a mesoscopic subset A ⊆ Γ n N [ω], we define its microscopic counterpart, (3.9) 3.2. The landscape near critical points. We now turn to the precise computation of the behavior of the measures Q β,N [ω](x) in the neighborhood of the critical points of F β,N [ω](x). We will see that this goes very much along the lines of the analysis in the one-dimensional case in Section 1. Let us begin by writing where (3.11) For y ∈ (−1, 1), these Z ℓ N can be expressed, using sharp large deviation estimates [10], as where o(1) goes to zero as |Λ ℓ | ↑ ∞. Note that as in the one-dimensional case, we identify functions on Γ n N with their natural extensions to R n . This means that we can express the right-hand side in (3.10) as (3.14) Here I N,ℓ [ω](y) is the Legendre-Fenchel transform of the log-moment generating function, We again analyze our functions near critical points, z * , of F β,N . Equations (3.10)-(3.15) imply: if z * is a critical point, then, for v ≤ N −1/2+δ , or, with z * = n j=1 z * ℓ , By standard properties of Legendre-Fenchel transforms, we have that I ′ N,ℓ (x) = U ′−1 N,ℓ (x), so that Summing over ℓ, we see that z * must satisfy the equation which nicely does not depend on our choice of the coarse graining (and hence on n). Finally, using that at a critical point , we get the explicit expression for the random numbersλ ℓ on the right hand side of (3.17) .

(3.22)
The determinant of the matrix A(z * ) has a simple expression of the form where o(1) ↓ 0, a.s., as N ↑ ∞. Combing these observations, we arrive at the following proposition.
Proposition 3.1. Let z * be a critical point of Q β,N . Then z * is given by (3.20) where z * is a solution of (3.21). Moreover, Proof. We only need to examine (3.13) at a critical point z * . The equation for the prefactor follows by combining (3.12) with (3.23). As for the exponential term, F β,N , notice that by convex duality Hence (3.14) equals Remark. The form given in Proposition 3.1 is highly suitable for our purposes as the dependence on n appears only in the denominator of the prefactor. We will see that this is just what we need to get a formula for capacities that is independent of the choice of the partition of I and has a limit as n ↑ ∞.
Eigenvalues of the Hessian. We now describe the eigenvalues of the Hessian matrix A(z * ).

) has at most one negative solution, and it has such a negative solution if and only if
Remark. To analyze the case when someλ k coincide is also not difficult. See [6].
Proof. To find the eigenvalues of A, just replaceλ k byλ k − γ in the first line of (3.23). This gives provided none of theλ ℓ − γ = 0. (3.27) is then just the demand that the first factor on the right of (3.29) vanishes. It is easy to see that, under the hypothesis of the lemma, this equation has n solutions, and that exactly one of them is negative under the hypothesis (3.28).
Topology of the landscape. From the analysis of the critical points of F β,N it follows that the landscape of this function is closely slaved to the one-dimensional landscape described in Section 1. We collect the following features: (i) Let m * 1 < z * 1 < m * 2 < z * 2 < · · · < z * k < m * k+1 be the sequence of minima resp. maxima of the one-dimensional function F β,N defined in (1.10). Then to each minimum, m * i , corresponds a minimum, m * i of F β,N , such that For any value m of the total magnetization, the function F β,N (x) takes its relative minimum on the set {y : y ℓ = m} at the pointx ∈ R n determined (coordinate-wise) by the equation Moreover, Remark. Note that the minimal energy curvesx(·) defined by (3.30) pass through the minima and saddle points, but are in general not the integral curves of the gradient flow connecting them. Note also that since we assume that random fields {h i (ω)} have bounded support, for every δ > 0 there exist two universal constants 0 < c 1 ≤ c 2 < ∞, such that

UPPER BOUNDS ON CAPACITIES
This and the next section are devoted to proving Theorem 1.3. In this section we derive upper bounds on capacities between two local minima. The procedure to obtain these bounds has two steps. First, we show that using test functions that only depend on the block variables m(σ), we can always get upper bounds in terms of a finite dimensional Dirichlet form. Second, we produce a good test function for this Dirichlet form. . Later we will be interested in pre-images of two minima of the function F β,N . We get the obvious upper bound Here To estimate such capacities it suffices to compute the capacity of some small set near the saddle point (see e.g. [3] or [8] for an explanation). For a given (small) In this section we will later choose For a fixed vector, v ∈ Γ n N , consider three disjoint subsets, We will compute the capacity of the Dirichlet form restricted to the set D N (ρ) with boundary conditions zero and one, respectively, on the sets W 1 ∩ D N (ρ) and W 2 ∩ D N (ρ). This will be done by exhibiting an approximately harmonic function with these boundary conditions. Before doing this, it will however be useful to slightly simplify the Dirichlet form we have to work with.
Cleaning of the Dirichlet form. One problem we are faced with in our setting is that the transition rates r N (x, x ′ ) are given in a somewhat unpleasant form. At the same time it would be nicer to be able to replace the measure Q β,N by the approximation given in (3.18). That we are allowed to do this follows from the simple assertion below, that is an immediate consequence of the positivity of the terms in the Dirichlet form, and of the Dirichlet principle.
Lemma 4.1. Let Φ N , Φ N be two Dirichlet forms defined on the same space, Γ, corresponding to the measure Q and transition rates r, respectively Q andr. Assume that, for all x, x ′ ∈ Γ, Then for any sets A, B . By the same token, The claimed relation follows.
To make use of this observation, we need to control the rates r N (x, x ′ ) and the measure Q β,N (x) in terms of suitable modified rates and measures. In fact, we see easily that so that, for all x ∈ D N (ρ) and for some K < ∞, it holds For that concerns the rates, let us first define, for σ ∈ S N , For all x ∈ Γ n N , we then have Notice that for all σ ∈ S N (x), |Λ − ℓ (σ)| is a constant just depending on x. Using that It follows easily that, for all x ∈ D N (ρ), With this in mind, we let L N be the generator of the dynamics on D N (ρ) with , and thus with reversible measure Q β,N (x). For u ∈ G A,B , we write the corresponding Dirichlet form as (4.17) 4.3. Approximately harmonic functions for Φ D N . We will now describe a function that we will show to be almost harmonic with respect to the Dirichlet form . . , n be the normalized eigenvectors of B, andγ i be the corresponding eigenvalues. We denote byγ 1 the unique negative eigenvalue of B, and characterize it in the following lemma.

Lemma 4.2. Let z * be a solution of the equation (3.21) and assume in addition that
Then, z * defined through (3.20) is a saddle point and the unique negative eigenvalue of B(z * ) is the unique negative solution,γ 1 ≡γ 1 (N, n), of the equation Moreover, we have that lim whereγ 1 is the unique negative solution of the equation Proof. The particular form of the matrix B allows to obtain a simple characterization of all eigenvalues and eigenvectors. The eigenvalue equations can be written as Assume for simplicity that all r kλk take distinct values. Then there is no non-trivial solution of these equation with γ = r kλk , and thus n ℓ=1 √ r ℓ u ℓ = 0. Thus,  Using (4.15) and noticing that Inserting the expressions for z * k /ρ k andλ k given by (3.20) and (3.22) into (4.26) and substituting the result into (4.25), we recover (4.20).
Since the left-hand side of (4.25) is monotone decreasing in γ as long as γ ≥ 0, it follows that there can be at most one negative solution of this equation, and such a solution exists if and only if left-hand side is larger than 1 for γ = 0. The claimed convergence property (4.21) follows easily.
We continue our construction defining the vectors v and the vectorsv (i) byv (4.28) We will single out the vectors v ≡ v (1) andv ≡v (1) . The important facts about these vectors is that (4.30) This implies the following non-orthogonal decomposition of the quadratic form A, (4.31) A consequence of the computation in the proof of Lemma 4.2, on which we shall rely in the sequel, is the following: There exists a positive constant δ > 0 such that independently of n, Proof. Due to our explicit computations, Consequently, the quantities φ k ≡ r kλk −γ 1 (N, n) are bounded away from zero and infinity, uniformly in N, n and k = 1, . . . , n. Since by (4.27) and (4.24) the entries of v are given by the assertion of the lemma follows.
Finally, define the function f : We claim that the function . Moreover, the following holds: Lemma 4.4. Let g be defined in (4.36). Then, for all x ∈ D N (ρ), there exists a constant c < ∞ such that Remark. The point of the estimate (4.37) is that it is by a factor ρ 2 smaller than what we would get for an arbitrary choice of the parameters v and γ 1 . We will actually use this estimate in the proof of the lower bound.
Proof. To simplify the notation we will assume throughout the proof that coordinates are chosen such that z * = 0. We also set A ≡ A(z * ). Using the detailed balance condition, we get Moreover, from the definition of Q β,N and using that we are near a critical point, we have that From (4.38) and (4.39), the generator can be written as Now we use the explicit form of g to obtain In particular, we get from here that Let us now insert these equations into (4.40): Using this fact, and collecting the leading order terms, we get Thus we will have proved the lemma provided that But note that from (4.31) we get that Hence using that by (4.28) r ℓ v ℓ =v ℓ and that by (4.30)v is orthogonal to v (j) with j ≥ 2, (4.46) follows and the lemma is proven.
Having established that g is a good approximation of the equilibrium potential in a neighborhood of z * , we can now use it to compute a good upper bound for the capacity. Fix now ρ = C ln N/N. Proposition 4.5. With the notation introduced above and for every n ∈ N, we get Here we used that ℓ r ℓ v 2 ℓ = ℓv 2 ℓ = 1. It remains to compute the sum over x. By a standard approximation of the sum by an integral we get  Inserting (4.50) into (4.49) we see that the left-hand side of (4.49) is equal to the right-hand side of (4.48) up to error terms.
It remains to show that the contributions from the sum outside D N in the Dirichlet form do not contribute significantly to the capacity. To do this, we define a global test function g given by Clearly, the only non-zero contributions to the Dirichlet form Φ N ( g) come from W 0 ≡ W 0 ∪ ∂W 0 , where ∂W 0 denotes the boundary of W 0 . Let us thus consider the sets W in the Dirichlet form of g restricted to W in 0 and to the part of its boundary contained in D N , i.e. to W in 0 ∩ D N , and by Φ W out 0 ( g) the Dirichlet form of g restricted to W out 0 . With this notation, we have The first term in (4.52) satisfies trivially the bound Performing the same computations as in (4.49) and (4.50) it is easy to show that Φ D ′ N (g) = Φ D N (g)(1 + o(1)), and then from (4.52) it follows that Φ (1)). (4.54) Consider now the second term in (4.52). Since g ≡ g on W 0 , we get where we also used that the function g has boundary conditions zero and one respectively on W 1 and W 2 . By symmetry, let us just consider the first sum in the r.h.s. of (4.55). For x ∈ ∂W in 0 ∩ W 1 it holds that (x, v) ≤ −ρ = −C ln N/N , and hence Using this bound together with inequality (4.41) to control (g(x + e ℓ ) − g(x)) 2 , we get for some constant c independent on N. The sum over x ∈ ∂W in 0 ∩ W 1 in the last term can then be computed as in (4.50). However, in this case the integration runs over the (n − 1)-dimensional hyperplane orthogonal to v and thus we have Inserting (4.58) in (4.57), and comparing the result with Φ D N (g), we get that the l.h.s of (4.57) is bounded as , which is positive if C is large enough. A similar bound can be obtained for the second sum in (4.55), so that we finally get (4.60) The last term to analyze is the Dirichlet form Φ W out 0 ( g). But it is easy to realize that this is negligible with respect to the leading term. Indeed, since for all x ∈ D c N it holds that F β,N (x) ≥ F β,N (z * ) + K ′ ln N/N, for some positive K ′ < ∞ depending on C, we get From (4.52) and the estimates given in (4.54), (4.59) and (4.61), we get that (1)) provides the claimed upper bound.
Combining this proposition with Proposition 3.1, yields, after some computations, the following more explicit representation of the upper bound. Corollary 4.6. With the same notation of Proposition 4.5,  To see this, note that where R is the diagonal matrix with elements R ℓ,k = δ k,ℓ √ r ℓ . Thus (4.64) as desired. Substituting in (4.48) the expression of Q β,N (z * ) given in Proposition (3.1), and after the cancellation due to (4.63), we obtain an upper bound which is almost in the form we want. The only n-dependent quantity is the eigenvalueγ 1 of the matrix B. Taking the limit of n → ∞ and using the second part of Lemma 4.2, we recover the assertion (4.62) of the corollary.
This corollary concludes the first part of the proof of Theorem 1.3. The second part, namely the construction of a matching lower bound, will be discussed in the next section.

LOWER BOUNDS ON CAPACITIES
In this section we will exploit the variational principle form Proposition 2.24 to derive lower bounds on capacities. Our task is to construct a suitable non-negative unit flow. This will be done in two steps. First we construct a good flow for the coarse grained Dirichlet form in the mesoscopic variables and then we use this to construct a flow on the microscopic variables. 5.1. Mesoscopic lower bound: The strategy. Let A and B be mesoscopic neighborhoods of two minima m A and m B of F β,N , exactly as in the preceding section, and let z * be the highest critical point of F β,N which lies between m A and m B . It would be convenient to pretend that m A , z * , m B ∈ Γ n N : In general we should substitute critical points by their closest approximations on the latter grid, but the proofs will not be sensitive to the corresponding corrections. Recall that the energy landscape around z * has been described in Subsection 3.2.
Recall that the mesoscopic capacity, Cap n N (A, B), is defined in (4.1). We will construct a unit flow, f A,B , from A to B of the form such that the associated Markov chain, P In view of the general lower bound (2.22), Eq. (5.2) implies that the mesoscopic capacities satisfy which is the lower bound we want to achieve on the mesoscopic level.
We shall channel all of the flow f A,B through a certain (mesoscopic) neighborhood G N of z * . Namely, our global flow, f A,B , in (5.1) will consist of three (matching) parts, f A , f and f B , where f A will be a flow from A to ∂G N , f will be a flow through G N , and f B will be a flow from ∂G N to B. We will recover (5.2) as a consequence of the three estimates whereas, (5.5) The construction of f through G N will be by far the most difficult part. It will rely crucially on Lemma 4.4. 5.2. Neighborhood G N . We chose again mesoscopic coordinates in such a way that z * = 0. Set ρ = N −1/2+δ and fix a (small) positive number, ν > 0. Define wherev ≡v (1) is defined in (4.28), and D N is the same as in (4.5). Note that in view of the discussion in Section 4, within the region G N we may work with the modified quantities, Q β,N and r ℓ ; ℓ = 1, . . . , n, defined in (4.11) and (4.17).
The boundary ∂G N of G N consists of three disjoint pieces, (5.7) We choose ν in (5.6) to be so small that there exists K > 0, such that uniformly over the remaining part of the boundary x ∈ ∂ r G N . Let g be the approximately harmonic function defined in (4.36) and (4.51). Proceeding along the lines of (4.49) and (4.50) we infer that, where I G N (x) ≡ {ℓ : x + e ℓ ∈ G N }. For functions, φ, on oriented edges, (x, x + e ℓ ), of D N , we use the notation φ ℓ (x) = φ(x, x + e ℓ ), and set .
In particular, the left hand side of (4.37) can be written as |dF [∇ g]|/ Q β,N (x). Let us sum by parts in (5.9). By (5.8) the contribution coming from ∂ r G N is negligible and, consequently, we have, up to a factor of order (1 + o(1)), (5.10) Furthermore, comparison between the claim of Lemma 4.4 and (4.49) (recall that ρ 2 = N 2δ−1 ≪ N −1/2 ) shows that the first term above is also negligible with respect to Φ N ( g). Hence, is a normalization constant. Let us fix 0 < ν 0 ≪ ν small enough and define, Thus, G 0 N is a narrow tube along the principalv-direction ( Figure 5.3). We want to construct φ in (5.12) such that the following properties holds: P1: f is confined to G N , it runs from ∂ A G N to ∂ B G N and it is a unit flow. That is, (5.14) uniformly in x ∈ G 0 N and ℓ = 1, . . . , n. P3: The flow f is negligible outside G 0 N in the following sense: For some κ > 0, Once we are able to construct f which satisfies P1-P3 above, the associated Markov chain P f N , X obviously satisfies (5.4). The most natural candidate for φ would seem to be ∇ g. However, since g is not strictly harmonic, this choice does not satisfies Kirchoff's law, and we would need to correct this by adding a (hopefully) small perturbation, which in principle can be constructed recursively. It turns out, however, to be more convenient to use as a starting choice From (3.16) and the decomposition (4.31) we see that In particular, there exists a constant χ 1 > 0 such that uniformly in x ∈ G N \ G 0 N and l = 1, . . . , n. Next, by inspection of the proof of Lemma 4.4, we see that there exists χ 2 , such that, uniformly in x ∈ G N and ℓ = 1, . . . , n. Notice that we are relying on the strict uniform (in n) positivity of the entries v ℓ , as stated in Lemma 4.3 Truncation of ∇g, confinement of f and property P1. Let C + be the positive cone spanned by the axis directions e 1 , . . . , e n . Note that the vectorv lies in the interior of C + . Define (see Figure 5.3) We assume that the constants ν and ν 0 in the definition of G N and, respectively, in the definition of G 0 N are tuned in such a way that G 2 Now we turn to the construction of the full flow. To this end we start by setting the values of φ ℓ on ∂ A G N equal to φ (0) if ℓ ∈ I G N (x) and zero otherwise. By (5.11) and the bound (5.20), the second of the relations in (5.14) is satisfied.
In order to satisfy Kirchoff's law inside G N , we write φ as φ = φ (0) + u with u satisfying the recursion, Since φ (0) ≡ 0 on G N \ G 1 N , we may trivially take u ≡ 0 on G N \ G 2 N and then solve (5.24) on G 2 N using the latter as an insulated boundary condition on ∂G 2 N ∩ G N . Interpolation of the flow inside G 2 N . We first solve (5.24) inside G 1 N . By construction, if x ∈ G 1 N then x − e ℓ ∈ G 1 N ∪ ∂ A G 1 N , for every ℓ = 1, . . . , n. Accordingly, let us slice G 1 N into layers L k as follows: Set and, for k = 0, 1, . . . , L j for all ℓ = 1, . . . , n .

(5.26)
Since all entries of v are positive, there exists χ 3 = c 3 (n) and M ≤ χ 3 /ρ, such that Now define recursively, for each x ∈ L k+1 , where the probability distribution, q 1 , . . . , q n , is defined as in (5.19). Obviously, this produces a solution of (5.24). The particular choice of the constants q ℓ in (5.19) leads to a rather miraculous looking cancellation we will encounter below. Properties P2 and P3. We now prove recursively a bound on u that will imply that Properties P2 and P3 hold. Let c k be constants such that, for all y ∈ L k , Then, for x ∈ L k+1 , we get by construction (5.28) and in view of (5.21) that By our choice of φ (0) in (5.23), However, for each i = 2, . . . , n, n j=1 (e j ,v)(e j ,v (i) ) = 0. (5.32) Therefore, with the choice q ℓ =v ℓ P kv k (1 + o(1)), we get uniformly in x ∈ G 1 N and l = 1, . . . , n. Thus, the coefficients c k satisfy the recursive bound c k+1 ≤ c k 1 + O(ρ 2 ) + χ 2 ρ 2 , (5.34) with c 0 = 0. Consequently, there exists a constant, c, such that c k ≤ kρ 2 ce kcρ 2 , (5.35) and hence, since M ≤ χ 3 /ρ, c M = O(ρ). As a result, we have constructed u on G 1 36) uniformly in x ∈ G 1 N and ℓ = 1, . . . , n. In particular, (5.15) holds uniformly in x ∈ G 1 N and hence, by (5.20), P3 is satisfied on G 1 N \ G 0 N . Moreover, since by construction φ ≡ 0 on G N \ G 2 N , P3 is trivially satisfied in the latter domain. Hence both P2 and P3 hold on Therefore, whatever we do in order to reconstruct φ, the total flow through G 2

Flows from
A to ∂ A G N and from ∂ B G N to B. Let f be the unit flow through G N constructed above. We need to construct a flow from A to ∂ A G N and, respectively, a flow from ∂ B G N to B, such that (5.5) holds and, of course, such that the concatenation f A,B = {f A , f, f B } complies with Kirchoff's law. We shall work out only the f A -case, the f B -case is completely analogous. The expressions for Φ N ( g) and Q β,N (x) appear on the right-hand sides of (4.48) and (3.13). For the rest we need only rough bounds: There exists a constant L = L(n), such that we are able to rewrite (5.38) as, This would imply a uniform stretched exponentially small upper bound on φ A at points x which are mesoscopically away from z * in the direction of ∇F β,N , for example for x satisfying With the above discussion in mind let us try to construct f A in such a way that it charges only bonds (x, y) for which (5.41) is satisfied. Actually we shall do much better and give a more or less explicit construction of the part of f A which flows through G 0 N : Namely, with each point x ∈ ∂ A G 0 N we shall associate a nearest neighbor path γ x = (γ x (−k A (x)), . . . , γ x (0)) on Γ n N such that (5.41) holds for all y ∈ γ x and, γ x (−k A (x)) ∈ A, γ x (0) = x and m(γ x (· + 1)) = m(γ x (·)) + 2/N. (5.42) The flow from A to ∂ A G 0 N will be then defined as By construction f A above satisfies the Kirchoff's law and matches with the flow f through G N on ∂ A G 0 N . Strictly speaking, we should also specify how one extends f on the remaining part ∂ A G N \ ∂ A G 0 N . But this is irrelevant: Whatever we do the P It remains, therefore, to construct the family of paths {γ x } such that (5.41) holds. Each such path γ x will be constructed as a concatenation γ x =γ ∪ η x . STEP 1 Construction ofγ. Pick δ such that δ − 1 < m A = m(m A ) and consider the partx[δ − 1, z * ] of the minimal energy curve as described in (3.30). Let γ be a nearest neighbor Γ n N -approximation ofx[δ − 1, z * ], which in addition satisfies m(γ(·+1)) = m(γ(·))+2/N. Since by (3.33) the curvex[δ−1, z * ] is coordinate-wise increasing, the Hausdorff distance betweenγ andx[δ − 1, z * ] is at most 2 √ n/N. Let x A be the first point where γ hits the set D N (ρ), and let u A be the last point where γ hits A (we assume now that the neighborhood A is sufficiently large so that u A is well defined). Thenγ is just the portion of γ from u A to x A . STEP 2 Construction of η x . At this stage we assume that the parameter ν in (5.6) is so small that G N lies deeply inside D N (ρ). In particular, we may assume that N , and, in view of (3.33), we may also assume that Therefore, x − x A has strictly positive entries and, as it now follows from (4.29), By construction G 0 N is a small tube in the direction ofv. Accordingly, we may assume that Ax, x − x A > 0 uniformly on ∂ A G 0 N . But this means that the function t : is strictly increasing. Therefore, F β,N is, up to negligible corrections, increasing on the straight line segment, [x A , x] ⊂ R n which connects x A and x. Then, our target path η x is a nearest neighbor Γ n N -approximation of [x A , x] which runs from x A to x . In view of the preceeding discussion it is possible to prepare η x in such a way that F β,N (z * ) − F β,N (·) > cN 2δ−1 along η x . Moreover, by (5.45) it is possible to ensure that the total magnetization is increasing along η x .
This concludes the construction of a flow f A,B satisfying 5.3.
In the sequel we shall index vertices of γ x =γ ∪ η x as, 47) for every y lying on the minimal energy curvex[δ − 1, z * ] and since the Hessian of F β,N is uniformly bounded onx[δ − 1, z * ], we conclude that if ν 0 is chosen small enough, then there exists c 2 > 0 such that uniformly in x ∈ ∂ A G 0 N . Finally, since the entries of v are uniformly strictly positive, it follows from (5.48) that, uniformly in x ∈ ∂ A and k ∈ {0, . . . , k A (x)}. Then the total microscopic flow, f A,B , can be decomposed as Therefore, (5.52) gives rise to the following decomposition of unity, where (P x , Σ) is the microscopic Markov chain from A to B which is associated to the flow f x . Consequently, our general lower bound (2.24) implies that We need to recover Φ N ( g) from the latter expression. In view of (5.1), write, Since we prove lower bounds, we may restrict attention to a subset of good realizations x of the mesoscopic chain X A,B whose P f A,B N -probability is close to one. In particular, (5.4) and (5.5) insure that the first term in the above product is precisely what we need. The remaining effort, therefore, is to find a judicious choice of f x such that the second factor in (5.56) is close to one. To this end we need some additional notation: Given a mesoscopic trajectory x = (x −ℓ A , . . . , x ℓ B ), define k = k(ℓ) as the direction of the increment of ℓ-th jump. That is, x ℓ+1 = x ℓ + e k . On the microscopic level such a transition corresponds to a flip of a spin from the Λ k slot. Thus, recalling the notation . By our choice of transition probabilities, p N , and their mesoscopic counterparts, r N , in (4.2), uniformly in ℓ and in all pairs of neighbors σ ℓ , σ ℓ+1 . Note that the cardinality, For x ∈ Γ n N , define the canonical measure, The second term in (5.56) is equal to If the magnetic fields, h, were constant on each set I k , then we could chose the flow f x (σ ℓ , σ ℓ+1 ) = µ x ℓ β,N (σ ℓ ) · 1/ Λ − k(ℓ) (σ ℓ ) , and consequently we would be done. In the general case of continuous distribution of h, this is not the case. However, since the fluctuations of h are bounded by 1/n, we can hope to construct f x in such a way that the ratio in (5.59) is kept very close to one.
Construction of f x . We construct now a Markov chain, P x , on microscopic trajectories, Σ = {σ 0 , . . . , σ ℓ B }, from S[x 0 ] to B, such that σ ℓ ∈ S[x ℓ ], for all ℓ = 0, . . . , ℓ B . The microscopic flow, f x , is then defined through the identity P The construction of a microscopic flow from A to S[x 0 ] is completely similar (it is just the reversal of the above) and we will omit it.
We now construct P x . STEP 1. Marginal distributions: For each ℓ = 0, . . . , ℓ B we use ν x ℓ to denote the marginal distribution of σ ℓ under P x . The measures ν x ℓ are concentrated on S[x ℓ ]. The initial measure, ν (5.60) STEP 2. Transition probabilities. The transition probabilities, q ℓ (σ ℓ , σ ℓ+1 ), in (5.60) are defined in the following way: As we have already remarked, all the microscopic jumps are of the form σ ℓ → θ + j σ ℓ , for some j ∈ Λ − k(ℓ) (σ), where θ + j flips the j-th spin from −1 to 1. For such a flip define Then the microscopic flow through an admissible bound, b = (σ ℓ , σ ℓ+1 ), is equal to Consequently, the expression in (5.59), and hence the second term in (5.56), is equal to This will imply that, which is the lower bound necessary to prove Theorem 1.3. The rest of the Section is devoted to the proof of (5.65). First of all we derive recursive estimates on Ψ ℓ for a given realization, x, of the mesoscopic chain. After that it will be obvious how to define T A,B . 5.6. Propagation of errors along microscopic paths. Let x be given. Notice that µ x ℓ β,N is the product measure, where µ β,N is the corresponding canonical measure on the mesoscopic slot S (j) On the other hand, according to (5.61), the big microscopic chain Σ splits into a direct product of n small microscopic chains, Σ (1) , . . . , Σ (n) , which independently evolve on S N . Therefore we are left with two separate tasks: On the microscopic level we need to control the propagation of errors along small chains and, on the mesoscopic level, we need to control the statistics of τ 1 [ℓ], . . . , τ n [ℓ]. The latter task is related to characterizing the set, T A,B , of good mesoscopic trajectories and it is relegated to Subsection 5.7 Small microscopic chains. It would be convenient to study the propagation of errors along small microscopic chains in the following slightly more general context: Fix 1 ≪ M ∈ N and 0 ≤ ǫ ≪ 1.
Back to the big microscopic chain. Going back to (5.68) we infer that the corrector of the big chain Σ satisfies the following upper bound: Let σ = (σ 0 , σ 1 , . . . ) be a trajectory of Σ (as sampled from P x ). Then, for every ℓ = 0, 1, . . . , ℓ B − 1, , . It is enough to control the first order approximation, The variables Y 1 , . . . , Y n are independent once x 0 is fixed. Thus, in view of our target, (5.65), we need to derive an upper bound of order (1 + O(ǫ)) for N -probability of order 1 − O(ǫ).

Good mesoscopic trajectories.
A look at (5.89) reveals what is to be expected from good mesoscopic trajectories. First of all, we may assume that it passes through the tube G 0 N (see (5.13)) of z * . In particular, x 0 ∈ G 0 N . Next, by our construction of the mesoscopic chain P f A,B N , and in view of (3.20) and (3.21), the step frequencies, τ j [ℓ]/ℓ, are, on average, proportional to ρ j . Therefore, there exists a constant, C 1 , such that, up to exponentially negligible P

holds.
A bound on microscopic moment-generating functions. We will now use the estimate (5.90) to obtain an upper bound on the product terms in (5.89). Clearly, B Now, for any t ≥ 0, ln µ where V x 0 (j),s β,N is the variance with respect to the tilted conditional measure, µ (5.93) However, µ x 0 (j),s β,N (·) is again a conditional product Bernoulli measure on S (j) .
(5.95) By (5.90) we need to consider only the case s/M j ≤ C 1 . Evidently, there exists δ 1 > 0, such that, On the other hand, since x 0 ∈ G 0 N , there exists δ 2 > 0, such that We use the following general covariance bound for product of Bernoulli measures, which can be derived from local limit results in a straightforward, albeit painful manner. Going back to (5.92) we infer from this that We say that a mesoscopic trajectory x = (x −ℓ A , . . . , x ℓ B ) is good, and write x ∈ T A,B , if it passes through G 0 N , satisfies (5.90) (and its analog for the reversed chain) and, in addition, it satisfies By construction (5.65) automatically holds for any x ∈ T A,B . Therefore, our target lower bound (5.66) on microscopic capacities will follow from Proof. By (5.49) we may assume that there exists C > 0 such that, for all x under consideration and for all ℓ = −ℓ A , . . . , ℓ B − 1, In view of (5.2) it is enough to check that -probabilities of order 1 − o(1). Fix δ > 0 small and split the sum on the left hand side of (5.103) into two sums corresponding to the terms with ℓ ≤ N 1/2−δ and ℓ > N 1/2−δ respectively. Clearly, uniformly in 0 ≤ ℓ ≤ N 1/2−δ . On the other hand, from our construction of the mesoscopic flow f A,B , namely from the choice (5.19) of transition rates inside G 0 N , and from the property (3.33) of the minimizing curvex(·), it follows that there exists a universal (ǫ-independent) constant, K < ∞, such that holds uniformly in ℓ > N 1/2−δ . A comparison with (5.102) yields (5.103).
The last proposition leads to the inequality (5.66), which, together the upper bound given in (4.62), concludes the proof of Theorem 1.3.

SHARP ESTIMATES ON THE MEAN HITTING TIMES
In this section we conclude the proof of Theorem 1.2. To do this we will use Equation (2.12) with A = S[m * 0 ] and B = S[M], where m * 0 is a local minimum of F β,N and M is the set of minima deeper than m * 0 . The denominator on the right-hand side of (2.12), the capacity, is controlled by Theorem 1.3. What we want to prove now is that the equilibrium potential, h A,B (σ), is close to one in the neighborhood of the starting set A, and so small elsewhere that the contributions from the sum over σ away from the valley containing the set A can be neglected. Note that this is not generally true but depends on the choice of sets A and B: the condition that all minima m of F β,N such that F β,N (m) < F β,N (m * 0 ) belong to the target set B is crucial.
In earlier work (see [4]) the standard way to estimate the equilibrium potential h A,B (σ) was to use the renewal inequality h A,B (σ) ≤ cap(A,σ) cap(B,σ) and bounds on capacities. This bound cannot be used here, since the capacities of single points are too small. We will therefore use another method to cope with this problem. We want to estimate the right-hand side of (6.1). This is expected to be of order Q β,N (m * 0 ), thus we can readily do away with all contributions where Q β,N is much smaller. More precisely, we choose δ > 0 in such a way that, for all N large enough, there is no critical point z of F β,N with F β,N (z) ∈ [F β,N (m * 0 ), F β,N (m * 0 ) + δ], and define U δ ≡ {m : F β,N (m) ≤ F β,N (m * 0 ) + δ}. (6.2) Denoting by U c δ the complement of U δ , we obviously have Lemma 6.1.
The main problem is to control the equilibrium potential h A,B (σ) for configurations σ ∈ S[U δ ]. To do that, first notice that where U δ (m) is the connected component of U δ containing m (see Fig. 6.1). Note that it can happen that U δ (m) = U δ (m ′ ) for two different minima m, m ′ ∈ M.
With this notation we have the following lemma. and The treatment of points (i) and (ii) is completely similar, as both rely on a rough estimate of the probabilities to leave the starting well before visiting its minimum, and it will be discussed in the next section.
Assuming Lemma 6.2, we can readily conclude the proof of Theorem 1.2. Indeed, using (6.5) together with (6.3), we obtain the upper bound where a(m * 0 ) is given in (1.19). On the other hand, using (6.6), we get the corresponding lower bound (1)).
6.2. Upper bounds on harmonic functions. We now prove Lemma 6.2 giving a detailed proof only for (i), the proof of (ii) being completely analogous. This requires, for the first time in this paper, to get an estimate on the minimizer of the Dirichlet form, the harmonic function h A,B (σ). First note that, since h A,B (σ) ≡ P σ (τ A < τ B ) for all σ / ∈ A ∪ B, the only non zero contributions to the sum in (i) come from those sets U δ (m) (at most two) whose corresponding m is such that there are no minima of M between m * 0 and m. By symmetry we can just analyze one of these two sets, denoted by U δ (m * ), assuming for definiteness that m * 0 < m * . Note also that since h A,B (σ) = 0 for all σ such that m * ≤ m(σ), the problem can be reduced further on to the set Define the mesoscopic counterpart of U − δ , namely, for fixed m * ∈ M and n ∈ N, let m * ∈ Γ n N be the minimum of F β,N (x) correspondent to m * , and define We write the boundary of U δ as Let max ℓ ρ ℓ ≪ θ(ε) ≪ 1, and for θ ≡ θ(ε) define As before, we denote by ∂G θ the boundary of G θ , and write Fig. 6.2). The strategy to control the equilibrium potential, P σ (τ A < τ B ), consists in estimating the probabilities , for σ ∈ G θ , in order to apply a renewal argument and to get from these estimates a bound on the probability of the original event.
Proceeding on this line, we state the following: Proposition 6.3. For any α ∈ (0, 1), there exists n 0 ∈ N, such that the inequality (6.14) holds for all σ ∈ S[U δ \ G θ ], n ≥ n 0 , and for all N sufficiently large.  We first observe that, for all σ ∈ S[U δ \ G θ ], The probability in the r.h.s. of (6.15) is the main object of investigation here. The idea which is beyond the proof of bound (6.14) is quite simple. Suppose that ψ is a bounded super-harmonic function defined on S[U δ \ G θ ], i.e.

20)
is super-harmonic in S[U δ \ G θ ] for all n ≥ n 0 and N sufficiently large.
The proof of Proposition 6.4 will involve computations with differences of the functions F β,N . We therefore first collect some elementary properties that we will use later. First we need some control on the second derivative of this function. From (3.14) we infer that Thus all the potential problems come from the function I N,ℓ . Lemma 6.5. For any y ∈ (−1, 1),
where O(1) is independent of N, n, and ν.
where C is a numerical constant independent of N, n, and ν.
The proof of this corollary is elementary and will not be detailed. The usefulness of (ii) results from the fact that |I ′ N,ℓ | is large on that domain. More precisely, we have the following lemma. Lemma 6.8. There exists ν > 0, independent of N and n, such that, if x ℓ /ρ ℓ > 1 − ν, then g ℓ (x) is strictly increasing in x ℓ and tends to +∞ as x ℓ /ρ ℓ ↑ +1; similarly if x ℓ /ρ ℓ < −1+ν, then g ℓ (x) is strictly decreasing in x ℓ and tends to −∞ as x ℓ /ρ ℓ ↓ −1.
Proof. Combine (ii) of Corollary 6.7 with Lemma 6.5 and note thath ℓ is bounded by hypothesis. The next step towards the proof of Proposition 6.4 is the following lemma. Lemma 6.9. Let m ∈ U δ \ G θ and denote by S(m) = {ℓ : m ℓ /ρ ℓ = 1}. Then there exists a constant c ≡ c(β, h) > 0, independent of N and n, such that the following holds. If Proof. From the relation I ′ N,ℓ (x) = U ′−1 N,ℓ (x), we get that, for all ℓ ∈ S(m), Here o(1) tends to zero as N → ∞.
We are concerned about small g ℓ (m). Subtracting 1 N i∈Λ ℓ tanh (β (m + h i )) on both sides of (6.36) and expanding the right-hand side to first order in g ℓ (m), and then summing over ℓ ∈ S(m) , we obtain by (1.20), non-zero derivative at m * . Moreover, by construction, m * is the only zero of this function in U − δ (m * ). From this observations, together with (6.37), we conclude that n ℓ=1 ρ ℓ g 2 ℓ (m) for some constant c < ∞. Here we used the triangle inequality and the fact that Under the hypothesis of the lemma, this gives the desired bound if |m − m * | ≥ c ′′ ε/ √ θ for some constant c ′′ < ∞. On the other hand, we can write, for ℓ ∈ S(m), Hence we get the bound   ℓ∈S(m) where in the last line we just used that m ∈ G θ . The inequalities (6.38) and (6.40) now yield (6.35), concluding the proof of the lemma.
We define the probabilities and observe that they are uniformly close to the mesoscopic rates defined in (4.2), namely for some c > 0 and ε = 1/n. Notice also that (6.45) With the above notation and using the convention 0/0 = 0, we get where we introduced the functions If x ℓ /ρ ℓ = ±1, the local generator takes the simpler form From Lemma 6.8 and inequalities (6.45), it follows that, for all ℓ such that (6.50) Let us now return to the case when x is not a boundary point. By the detailed balance conditions, it holds that which implies, together with (6.44), Inserting the last bounds in (6.47) and (6.48), and with some computations, we obtain where ∓ ≡ −sign (g ℓ (x)) = −sign (g ℓ (x − e ℓ )).
The sum of the probabilities in the r.h.s. of (6.92) corresponds to the quantity P σ(t−1) st,ℓt defined in (6.43). From the inequalities (6.44) and (4.15), it follows that, for some constant c > 0 depending on β and on the distribution of the field, where the second inequality follows by our choice of the path m. Now, since |Λ ± ℓ (m * )|/N = 1 2 (ρ ℓ ± m * ℓ ), using the expression (3.20) for m * ℓt and continuing from (6.93), we obtain P σ(t−1) st,ℓt ≥ c ′ ρ ℓt . (6.94) Inserting the last inequality in (6.92), and using that, by definition of the path m, the number of steps corresponding to a spin-flip in Λ ℓ is equal to |v ℓ |N, for all ℓ = {1, . . . , n} , we get where in the third line we used the inequality v 1 ≤ ε −1/2 v 2 ≤ √ ε, and in the last line we used that m(0) = m * + v ∈ G θ . By our choice of θ ≫ ε, there exists n 0 ∈ N such that, for all n ≥ n 0 , ε θ − √ ε ln(c ′ ) ≤ βδ 2 . For such n's, inequality (6.95) yields the bound (6.89) and concludes the proof of the Lemma.
We finally state the following proposition: Proof. Let us first consider a configuration σ ∈ S[∂ A G θ ]. Then it holds where in the second line we applied the Markov property, and in the last line we insert the result (6.12). Taking the maximum over σ ∈ S[∂ A G θ ] on both sides of (6.97), and rearranging the summation, we get where in the last line we used the bound (6.88). This concludes the proof of (6.96) for σ ∈ S[∂ A G θ ].
Then, let us consider σ ∈ S[U δ \ ∂ A G θ ]. As before, it holds , and exponentially small in N for all σ ∈ S[U δ \ G θ ] (due to Proposition 6.3). Inserting the bound (6.98) in the last equation, provides Eq. (6.96) for σ ∈ S[U δ \ ∂ A G θ ] and concludes the proof.
The proof of formula (6.5) now follows straightforwardly. From (6.96), we get where in the second inequality we used the expression (1.9) for Q β,N (m * 0 ), while in the last line we applied the bound F β,N (m) ≤ F β,N (m * ) = F β,N (m * ), and then bounded the cardinality of U δ by N n . Finally, choosing α small enough, namely we can easily ensure that (6.100) implies (6.5). In exactly the same way one proves (6.6). This concludes the proof of Lemma 6.2 and thus of Theorem 1.2.