Metastability for Glauber dynamics on the complete graph with coupling disorder

Consider the complete graph on $n$ vertices. To each vertex assign an Ising spin that can take the values $-1$ or $+1$. Each spin $i \in [n]=\{1,2,\dots, n\}$ interacts with a magnetic field $h \in [0,\infty)$, while each pair of spins $i,j \in [n]$ interact with each other at coupling strength $n^{-1} J(i)J(j)$, where $J=(J(i))_{i \in [n]}$ are i.i.d. non-negative random variables drawn from a probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature $\beta \in (0,\infty)$. We show that there are critical thresholds $\beta_c$ and $h_c(\beta)$ such that, in the limit as $n\to\infty$, the system exhibits metastable behaviour if and only if $\beta \in (\beta_c, \infty)$ and $h \in [0,h_c(\beta))$. Our main result is a sharp asymptotics, up to a multiplicative error $1+o_n(1)$, of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of $J$, while the correction terms do. The leading order of the correction term is $\sqrt{n}$ times a centred Gaussian random variable with a complicated variance depending on $\beta,h$, on the law of $J$ and on the metastable state. The critical thresholds $\beta_c$ and $h_c(\beta)$ depend on the law of $J$, and so does the number of metastable states. We derive an explicit formula for $\beta_c$ and identify some properties of $\beta \mapsto h_c(\beta)$. Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.


Introduction and main results
1.1.Background.Interacting particle systems evolving according to a Metropolis dynamics associated with an energy functional called the Hamiltonian, may be trapped for a long time near a state that is a local minimum of the free energy, but not a global minimum.The deepest local minima are called metastable states, the global minimum is called the stable state.The transition from a metastable state to the stable state marks the relaxation of the system to equilibrium.To describe this relaxation, one needs to identify the set of critical configurations the system must attain in order to achieve this transition and to compute the crossover time.These critical configurations correspond to saddle points in the free energy landscape.
Metastability for interacting particle systems on lattices has been studied intensively in the past.For a summary, we refer the reader to the monographs by Olivieri and Vares [13], and Bovier and den Hollander [6].Successful attempts towards understanding metastable behaviour in random environments were made for the random field Curie-Weiss model, by Mathieu and Picco [12], Bovier, Eckhoff, Gayrard and Klein [3] and Bianchi, Bovier and Ioffe [1,2].Recently, there has been interest in metastability for interacting particle systems on random graphs.This is challenging, because the crossover times typically depend on the realisation of the graph.In den Hollander and Jovanovksi [11] and Bovier, Marello and Pulvirenti [7], Glauber dynamics on dense Erdős-Rényi random graphs was analysed.Earlier work on metastability for Glauber dynamics on sparse random graphs can be found in Dommers [8] (random regular graph) and Dommers, den Hollander, Jovanovski and Nardi [10] (configuration model).The present paper is a first step towards the study of metastability for Glauber dynamics on Chung-Lu-like random graphs.
To the best of our knowledge, Tindemans and Capel [14] and Dommers, Giardinà, Giberti, van der Hofstad and Prioriello [9] are the only references where the model with the interaction Hamiltonian in (1.2) below has been studied in detail.Both focus on equilibrium properties only.
1.2.Glauber dynamics on the complete graph with coupling disorder.Let K n be the complete graph on n vertices.Each vertex carries an Ising spin that can take the values −1 or +1.Let S n = {−1, +1} [n] denote the set of spin configurations on K n , where [n] = {1, 2, . . ., n}.Let (Ω, F, P) be an abstract probability space, and let J = (J(i)) i∈[n] be a sequence of i.i.d.random variables on this probability space taking values in a finite set {a 1 , . . ., a k } ⊂ [0, ∞) of cardinality k ∈ N. The distribution of these random variables is given by (1.1) with ∈[k] ω = 1.Let H n : S n → R be the interaction Hamiltonian defined by where h ∈ [0, ∞) is the magnetic field.We consider Glauber dynamics on S n , defined as the continuous-time Markov process with transition rates where β ∈ (0, ∞) is the inverse temperature, σ ∼ σ means that σ differs from σ by a single spin-flip and [•] + is the positive part.This dynamics is reversible with respect to the Gibbs measure where the normalising constant Z n is called the partition sum.Note that the reference measure for (1.4) is the counting measure on S n .We write to denote a path of the Glauber dynamics on S n , and P σ and E σ to denote probability and expectation on path space given σ 0 = σ (we suppress J, h, β and n from the notation).For fixed n, if h = 0 the Hamiltonian in (1.2) has two global minima, at σ ≡ +1 and σ ≡ −1, while if h > 0 it achieves a global minimum at σ ≡ +1 and a local minimum at σ ≡ −1.The latter is the deepest local minimum not equal to the global minimum (at least for h small enough).However, in the limit as n → ∞, these do not form a metastable pair of configurations because entropy comes into play.
1.3.Metastability on the complete graph with coupling disorder.In this section we state our main results.1.3.1.Empirical magnetisations.The relevant quantity to monitor in order to characterise the metastable behaviour is the disorder weighted magnetisation The following quantities will be essential for coarse-graining.Define the level sets and the level magnetisations and note that σ) depends on σ only through m n (σ).Thus, with abuse of notation, we may define (1.10) 1.3.2.Thermodynamic limit.As n → ∞, by the law of large numbers the random function K n converges uniformly in probability to a deterministic function K given by Similarly, the random free energy function F n converges uniformly in probability to a deterministic function F β,h (see (2.15) and (2.26) below for explicit formulas).In Section 3, we show that the stationary points of F β,h are given by m = (m ) ∈[k] , where Note that, via (1.12), the k-dimensional vector m is fully determined by the real number K(m).Therefore, finding the stationary points of F β,h reduces to finding the solutions of the equation Namely, if β ∈ (0, β c ], then the system is not in the metastable regime for any h ∈ [0, ∞), while if β ∈ (β c , ∞), then, for h ∈ [0, ∞) small enough, it is in the metastable regime (i.e., (1.13) has more than one solution at which T β,h is not tangent to the diagonal).Given β ∈ (β c , ∞), the critical magnetic field h c (β) is the minimal value of h for which the system is not metastable.The metastable regime is thus In Section 3, we show that β → h c (β) is continuous on (β c , ∞), with (1.16) where the explicit value of C is given in (3.12) below.Interestingly, β → h c (β) is not necessarily monotone, i.e., the metastable crossover may be re-entrant.
It turns out that there exists an ∈ [k] (depending on β, h and on the law of the components of J), such that F β,h has 2 + 1 stationary points.
Let G(A, B) be the gate between two disjoint subsets A and B of M n .We refer to [6, Section 10.1] for a precise definition of the gate.Fix m n ∈ M n as the initial magnetisation.Throughout the paper we assume that the following hypotheses hold for m n .
(2) The Hessian of F n has only non-zero eigenvalues at m n and at all the points in G(m n , M n (m n )).
(3) There is a unique point t n in G(m n , M n (m n )), which will often be called simply saddle point.
where r is defined in (4.9) below.
Hypothesis 1(2) and ( 3) are made to avoid complications.Hypothesis 1(4) is needed in the proof of Lemma 4.1 below (as in [6,Lemma 14.9]).Neither is very restrictive: if for some parameter choice they fail, then after an infinitesimal parameter change they hold.Moreover, if Hypothesis 1(3) fails, it is sufficient to compute separately the contribution to the crossover time of the various saddle points in the gate.
Let S n [m n ] and S n [M n (m n )] denote the sets of configurations in S n for which the level magnetisations are m n and are contained in M n (m n ), respectively.For A ⊂ S n , write to denote the first hitting time or return time of A.
We next state our main results for the crossover time.Theorem 1.1 provides a sharp asymptotics for the average crossover time from any metastable state to the set of states with lower free energy.Theorem 1.2 shows that asymptotically the crossover time is exponential on the scale of its mean, a property that is standard for metastable behaviour.

Theorem 1.1 (Average crossover time with coupling disorder).
Let A n (•) be the k×k Hessian matrix defined in (4.2) below, and γ n the unique negative solution of the equation in (4.20) below.For every m n ∈ M n satisfying Hypothesis 1 and within the metastable regime (1.15), uniformly in σ ∈ S n [m n ], and with P-probability tending to 1, e βn[Fn(tn)−Fn(mn)] .(1.19) Theorem 1.2 (Exponential law with coupling disorder).For every m n ∈ M n satisfying Hypothesis 1 and within the metastable regime (1.15), uniformly in σ ∈ S n [m n ] and with P-probability tending to 1, As the average crossover time estimated in Theorem 1.1 is a random variable, we next provide more information on the randomness of the quantity in the right-hand side of (1.19), which depends on the realisation of the random variable J.The prefactor in (1.19) converges with P-probability tending to 1 to a deterministic limit, which depends on the law of J but not on the realisation of J.However, the exponent does not converge to a deterministic limit.In Theorem 1.3 we compute the exponent up to order O(1).Recall that For every m n ∈ M n satisfying Hypothesis 1 and within the metastable regime (1.15), in distribution, where Z is a normal random variable with mean zero and variance in (0, ∞), defined on (Ω, F, P) and independent of J.
The variance of Z turns out to be a complicated function of β, h and the distribution of J.We refer to Section 6.In the present paper we consider pair interaction random variables with a discrete distribution only.It seems hard to obtain results with a similar precision for continuous distributions.The techniques employed in [1,2] do not carry over, because the error introduced by the coarse-graining turns out to be quadratic rather than linear.
More specifically, we first find a sharp approximation of the Dirichlet form associated with the coarse-grained dynamics.We use these results, together with lumpability properties and well-known variational principles, to obtain sharp capacity estimates that are key quantities in the proof.For a more detailed overview of the methods, we refer the reader to the monograph by Bovier and den Hollander [6].The remainder of the paper is organised as follows.Section 2 provides quantities and notations that are needed throughout the paper.Section 3 identifies the metastable regime.Section 4 provides a sharp approximation of the Dirichlet form associated with the Glauber dynamics in the presence of the disorder.Section 5 provides estimates on capacity and on the metastable valley measure.Section 6 proves Theorems 1.1-1.3.Appendix A contains a brief overview on known results for the standard CW model, which corresponds to the setting without disorder.Appendix B gives numerical evidence for the presence of multiple metastable states for suitable choices of β, h and of the law of the components of J. Appendix C contains an example in which β → h c (β) is not increasing, implying the possibility of a re-entrant metastable crossover.Appendix D provides the limit as n → ∞ of the prefactor in (1.19).

Preparations
Section 2.1 introduces further notation and writes the Hamiltonian in terms of the level magnetisations.Section 2.2 introduces the Dirichlet form associated with the Glauber dynamics and rewrites this in terms of the level magnetisations.Section 2.3 computes gradients and Hessians of the free energy as a function of the level magnetisations.Section 2.4 closes with an approximation of the free energy that will be needed later on.
Hence m n (σ) takes values in the set (2.3) The configurations corresponding to M ⊆ Γ n are denoted by For singletons M = {m} we write which is the Hamiltonian in (1.2), except for the diagonal term − 1 2n i∈[n] J 2 (i), which is a constant shift.Using the notation above, we can write the Hamiltonian in (2.5) as a ω ,n m ,n (σ) where we abbreviate 2.2.Dirichlet form and mesoscopic dynamics.By (1.3)-(1.4), the Dirichlet form associated with the Glauber dynamics equals where h is a test function on S n taking values in [0, 1].Because of (2.6), for any h such that h(σ) = h(m n (σ)), with h a test function on Γ n , we have (2.9) For m, m ∈ Γ n , we write m ∼ m when there exists an Moreover, for ∈ [k] and σ ∈ S n with m n (σ) = m, the cardinality of the set {σ ∈ S n : σ ∼ σ, m n (σ ) = m ,± } equals 1∓m 2 |A ,n |, namely, the number of (∓1)-spins in σ with index in A ,n .Furthermore, (2.12) |{σ ∈ S n : as is seen by counting the number of (−1)-spins with label in A ,n of a configuration with -th level magnetisation m .Using these observations, we can rewrite (2.9) as (2.13) and put (2.15) where E n (m) is defined in (2.7).Moreover, define (2.16) With this notation, we can write the mesoscopic measure and so the Dirichlet form in (2.13) becomes 2.3.Gradients and Hessians.Denote the Cramér entropy by

Define
(2.20) where the error term is Recalling (2.7), we compute ω I C (m ), (2.26) which corresponds to the uniform limit in probability of (2.28) The same formulas apply for I n , F n , with an error term O(n −1 ).

Additional computation.
We conclude with a computation that will be useful later on.Recalling (2.11), we write where The same formula applies for I n with an error term of order O(n −1 ), and hence Note that ∆ ± ,n = O(1).Therefore, using (2.15), we get (2.32)

Metastable regime
Section 3.1 identifies the stationary points of Fn .Section 3.2 identifies the metastable regime.Section 3.3 provides details on the 1-dimensional metastable landscape.

Stationary points of Fn and F
2) can be rewritten as Similarly, the critical points m = (m ) ∈[k] of F β,h solve the deterministic equation Note that this can also be obtained directly from (3.3) after replacing ω ,n by its mean value ω .
3.2.Metastable regime.We are interested in identifying the metastable regime, i.e., the set of pairs (β, h) for which F β,h has more than one minimum.Put a ω m .
From the characterisation of the critical points of F β,h in (3.4) it follows that Consequently, the problem of solving the k-dimensional system in (3.4) can be reduced to solving the 1-dimensional equation (3.6).Recalling Hypothesis 1(2), the system is in the metastable regime if and only if (3.6) has more than one solution that is not tangent to the diagonal.Compute (3.7) For h = 0, the system is metastable when , in which case T β,h has a unique inflection point at K = 0, implying that (3.6) has three solutions K ∈ {−K * , 0, +K * } with K * > 0. Otherwise (3.6) has only one solution K = 0.
We proceed with the more interesting case h > 0.
Proof.For h > 0 and K positive and large enough, T β,h (K) < 0.Moreover, for h > 0 and K negative with |K| large enough, T β,h (K) > 0. Therefore, T β,h has at least one inflection point and that the number of inflection points of T β,h cannot be even: it takes values in {1, 3, . . ., 2k − 1} depending on β, h and the law of the components of J. Consequently, if ) is the number of inflection points, then the cardinality of the solutions of (3.6) takes values in {1, 3, . . ., 2 + 1} depending on β, h and on the law of the components of J.
We conjecture that for any finite k there exist β, h and a law of the components of J such that (3.6) has any number of solutions in the set {1, 3, . . ., 2k + 1}.We found numerical evidence for this fact for k ∈ {2, 3, 4}.See Appendix B. Lemma 3.2 (Unique strictly positive solution).For every β > 0 and h > 0, (3.6) has exactly one strictly positive solution.
Proof.Put W (K) = T β,h (K) − K.The solutions of (3.6) are the roots of W . Clearly, a ω > 0 is finite.Therefore, by continuity, a root of W (K) exists in (0, ∞).
Let K be the smallest positive root of W . Next we will prove that this root is unique.Indeed, W (K) < 0 when K ∈ [0, ∞), meaning that K → W (K) is strictly decreasing.By continuity, since W (K) > 0 for all K ∈ [0, K), we have W ( K) ≤ 0 and lim K→∞ W (K) = −1.Therefore, W (K) < 0 for all K ∈ ( K, ∞), and so W is strictly decreasing.Moreover, W (K) < W ( K) = 0 for all K ∈ ( K, ∞).Thus, K is the only positive root of W .
(3.6) has at least three solutions not tangent to the diagonal if and only if there exists Proof.Using Lemma 3.2, we see that (3.6) has at least three solutions if and only if it has at least two strictly negative solutions.As above, we define W (K) = T β,h (K) − K.The solutions of (3.6) are the roots of W . Now, assume that there exists a K < 0 such that K > T β,h ( K).Since W ( K) < 0 and W (0) > 0, W (K) has a root in ( K, 0), implying that (3.6) has at least one solution in ( K, 0).Moreover, since With the same argument it can be shown that the negative roots of W are always even.The opposite implication is trivial.Remark 3.1.Applying the intermediate value theorem to the derivative of W (K) = T β,h (K) − K, we get that if the condition in Lemma 3.3 is satisfied, then there exists a K < 0 such that T β,h ( K) = 1 and K > T β,h ( K). ♠ Theorem 3.4 (Metastable regime).Define, as in (1.14), (3.10) The metastable regime is where the minimum is over all ∈ [k] such that the quantity between brackets is positive.
Proof.Recalling Lemma 3.3, we look for conditions for the existence of a K < 0 satisfying (3.9).If such a K exists, then by Remark 3.1 there exists a K < 0 satisfying (3.9) such that T β,h ( K) = 1, which reads Since the left-hand side of (3.13) is positive, it admits solutions only if Therefore, (3.14) is a necessary condition for the metastable regime.Now assume (3.14).Since tanh x ∼ x, x → 0, for |K| β(max ∈[k] a ) −1 and h ↓ 0, we have and proves the existence of a negative solution.A positive solution is guaranteed by Lemma 3.2.The existence of a third (strictly negative) solution of (3.4), for every β > β c and for h ↓ 0, follows as in the proof of Lemma 3.3.Therefore, the lower bound on β c is sharp.Since h → T β,h (K) is strictly increasing for every fixed β > 0 and K ∈ R, there exists a unique critical curve β → h c (β) such that the system is metastable for 0 ≤ h < h c (β) and not metastable for h ≥ h c (β).We know that h c (β) > 0 for β > β c .By passing to the parametrisation g = hβ, we get that β → T β,g (K) is strictly decreasing for every g and for every K < 0, from which it follows that β → g c (β) = βh c (β) is non-decreasing.
We next focus on the limit of h c (β) as β → ∞.By Lemma 3.3, we may focus on the existence of K satisfying (3.9).In the limit as and, for all ∈ [k], where we set − h a 0 = −∞ and − h a k+1 = ∞.Thus, for K ∈ − h a −1 , − h a , (3.9) can be written as Therefore, (3.9) has a solution if and only if there exists an in which case a solution K of (3.9) exists in (3.21) Note that the quantity between brackets in (3.12) is always positive for = 1.Thus, the minimum is always finite.
The proof is complete after we show why we may drop the case where K = − h a for some ∈ [k].In this case the condition for K to satisfy (3.9) is which implies (3.20).Thus, if K = −h a satisfies (3.9), then also some other K in (3.21) satisfies (3.9).Therefore, the condition in (3.20) is equivalent to having metastability.Lemma 3.5 (Re-entrant crossover).The function β → h c (β) is not necessarily nondecreasing.
Proof.In Appendix C we provide an example of β → h c (β) that is not increasing.

Bounds on the inflection points and on the critical curve.
Lemma 3.6 (Bounds on inflection points).All solutions of T β,h (K) = 0 are contained in the interval In particular, they are all strictly negative. Proof.
Proof.Use Lemma 3.3 to characterise the metastable regime and Remark 3.1.We claim that if a solution K of (3.9) with T β,h ( K) = 1 exists, then it must be negative and strictly less than an inflection point.Using this fact, together with Lemma 3.6 and the inequality in (3.9), we obtain a necessary upper bound on h: Using that tanh(β[a K + h]) > −1, we conclude the proof.
We are left to prove the claim.By Lemma 3.6, all inflection points are negative, and which contradicts what we have proved for all K ∈ ( K, ∞).Recalling definitions (2.25) and (3.5), using Lagrange multipliers and integrating the condition K n (m) = K, we obtain Lemma 3.8 (Alternative characterisation for the critical points). ( Proof.Similar to [3,Lemma 7.4]. We have already seen that K n (m) fully determines any critical value m of F n , and is useful to order them.Lemma 3.8 exhibits the one-dimensional structure underlying the metastable landscape and provides a tool to describe the nature of the critical points of F n .
Remark 3.2.The above results extend to the limit n → ∞: replace F n by F β,h and G n by G β, , obtained after replacing ω ,n by ω in (3.27), and K n (•) by K(•).♠

Approximation of the Dirichlet form near the saddle point
In this section we approximate the Dirichlet form associated with the coarse-grained dynamics near the saddle point.This is a key step to obtain capacity estimates in the following section.Further details and examples on the techniques we use here can be found in [6, Chapters 9, 10 and 14].
Section 4.1 introduces some key quantities that are needed to express the mesoscopic measure.Section 4.2 introduces an approximate mesoscopic measure that leads to an approximate dynamics.Section 4.3 approximates the harmonic functions associated with this dynamics.Section 4.4 computes an approximate Dirichlet form.Section 4.5 uses the latter to approximate the full Dirichlet form.

Key quantities. Let m
where d is the Euclidean distance and C ∈ (0, ∞) is a constant.Abbreviate the Hessian of and put (4.3) By (2.28), (4.4) Note that A n (m) is a diagonal matrix minus a rank one matrix.Compute For m ∈ D n , define and where rn is defined in (2.16).The transition rates rn define a random dynamics on D n that is reversible with respect to the mesoscopic measure Qn .The corresponding Dirichlet form is where u is a test function on D n .Put (4.9) r = rn m, m ,+ = rn t n , t ,+ n .
where A n is defined in (4.3).Note that n denote the unique negative eigenvalue of B n , and v the corresponding unitary eigenvector.Define As in [6,Remark 10.4], it follows by Hypothesis 1 that A n has all strictly positive eigenvalues but one strictly negative.It can be seen that the same property holds for the eigenvalues of B n .♠ Lemma 4.1 (Eigenvalue).The eigenvalue γ n is the unique solution of the equation Proof.We follow the line of proof of [6, Lemma 14.9], using the last point in Hypothesis 1.In our case, [6, Eq. (14.7.12)] reads (4.21) As in [6, Lemma 14.9], since the left-hand side of (4.20) is increasing in γ n for γ n ≥ 0, a negative solution of (4.20) exists if and only if Using Recall the definition of M n (m n ) given in (1.17).Let W 0 be a strip in Γ n of width Cn −1/2 log 1/2 n such that t n ∈ W 0 , M n (m n ) ∩ W 0 is empty and W c 0 consists in two non-neighbouring parts: W 1 containing m n and W 2 containing M n (m n ).Moreover, we require that, for some fixed constant c > 1, By choosing W 0 and D n suitably we have, for m ∼ m (i.e., rn (m, m ) > 0) and c ∈ (0, ∞) large enough (coming from the definition of W 0 ), (4.28) Recalling (4.8)-(4.9),we have  1)] for all m ∈ W c 0 ∩ D n .The latter can be proved by approximating the Gaussian integral by 0 or 1 when v, m − t n is proportional to −n −1/2 log 1/2 n or n −1/2 log 1/2 n, respectively.♠ 4.5.Final Dirichlet form approximation.We are now ready to compare E Sn with ẼDn .Let h : Then, using (4.25)-(4.27),we obtain Using (4.14) and (4.16), we obtain where the third equality follows from (4.25)-(4.27)together with (4.14), and the last equality follows from (4.29)-(4.30).

Capacity and valley estimates
Section 5.1 provides sharp asymptotic upper bounds and lower bounds on the capacity of the metastable pair between which the crossover is being considered.These estimates use the results of the Section 4 together with the Dirichlet principle and the Berman-Konsowa principle, which are variational representations of capacity.Section 5.2 provides a sharp asymptotic estimate for the mesoscopic measure of the valleys of the minima of F n , which leads to a sharp asymptotic estimate for F n inside this valley.5.1.Capacity estimates.Given a Markov process (x t ) t≥0 with state space S, a key quantity in the potential-theoretic approach to metastability is the capacity cap(A, B) of two disjoint subsets A, B of S. This is defined by (see [6, Eq. (7.1.39)]) where µ is the invariant measure and P x is the probability distribution of the Markov process starting in x.
Recall that M n is the set of local minima of F n .
Remark 5.1.Proposition 5.1 holds for any subset M n ⊆ M n \m n , separated from m n by t n , independently on the values of F n on M n .♠ 5.1.1.Upper bound: Dirichlet principle.An important characterisation of the capacity between two disjoint sets is given by the Dirichlet principle.For our quantity of interest this states that where H is the set of functions from S n to [0, 1] that are equal to Given that, by assumption, G(m n , M n ) = {t n }, we use the Dirichlet principle in (5.3) to obtain an upper bound on the capacity.We take as test function h ∈ H defined in Section 4.5 and, using (4.32), we obtain where cap Γ denotes the capacity for the process (m n (σ t )) t≥0 , i.e., the projection of the process (σ t ) t≥0 on the magnetisation space Γ n .We write P Γ and E Γ to denote the law of (m n (σ t )) t≥0 induced by the law P of (σ t ) t≥0 , and its expectation, respectively.By the lumpability, we can focus on the dynamics on Γ n .Following the line of argument in [6, Section 10.3] (with ε = 2 n and d = k), we obtain the lower bound where we use (4.29) and (4.30).
We sketch the proof.The main idea is to use the Berman-Konsowa principle for a suitable defective flow.More precisely, given disjoint subsets A, B of the state space, for any defective loop-free unit flow f A,B from A to B with defect function δ (as defined in [6, Definition 9.2]), we can estimate (see [6,Lemma 9.4], and notation therein) where [•] + denotes the positive part and γ is a self-avoiding path from A to B. It turns out that, with a suitable choice of the flow f , the product in the right-hand side of (5.7) is bounded from below by 1 + O( n −1/2 log 1/2 n), and the sum over γ from below by ẼDn (g, g)[1 + o n (1)].This proves (5.6).
We give a sketch of the test flow definition in our setting.Here A = {m n } and B = M n .Let v * be the eigenvector corresponding to the unique negative eigenvalue of the Hessian of F n at the saddle point t n (unique gate point in G({m n }, M n )).Let G n be the cylinder in R k intersected with Γ n , centred at t n , with axis v * , radius ρ = C n −1/2 log 1/2 n and length ρ = C n −1/2 log 1/2 n.We will denote by ∂ B G n the base facing B and by ∂ A G n the central part of radius C n −1/2 log 1/2 n of the base facing A, with C < C. Choose the constants so that G n is contained in D n defined in (4.1).
We define a defective flow f A,B from A to B consisting of three parts: where g is defined in (4.24), Qn in (4.6), r in (4.9) and (5.9) The contribution to the sum in brackets in (5.7) turns out to be negligible outside G n .Therefore, no further conditions on the flows f A and f B are necessary, provided the total flow out of A is 1 and the total flow f A,B is defective and loop-free.5.2.Measure of the valley.In order to prove Theorem 1.1, we need the following estimate on the measure of the valley of the minima of F n .For m n ∈ M n , let A(m n ) ⊂ Γ n be the valley of m n as defined in [6,Eq. (8.2.10)].
Lemma 5.2 (Gibbs weight of the valley).Given m n ∈ M n , (5.10) where Q n is the mesoscopic measure defined in (2.17), and A n (m n ) is the k × k Hessian matrix defined in (4.2).
Proof.The proof follows that of [6, Lemma 10.12 and (10. where c is a constant.The condition y ∈ A(m n ) is needed to ensure that F n (y) > F n (m n ), implying that c is positive.Therefore, we obtain the rough estimate (5.12) where we use that |Γ n | ≤ n k .The bound in (5.12) is sufficient to show that where we use the Taylor expansion (5.14) and the approximation of the sum by an integral is correct up to an error 1+O(ρ).In the last lines we approximated the Gaussian integral on intervals [−ρ, ρ] by the Gaussian integral on R, with an error 1 + O(n −c ).We conclude by looking at (5.12) and (5.13), and noting that for

Proof of the theorems
In this section we prove Theorems 1.1-1.3.Section 6.1 uses the asymptotics for the capacity of the metastable pair from Section 5.1 and the asymptotics for the mesoscopic measure from Section 5.2 to prove Theorem 1.1.Section 6.2 proves Theorem 1.2.Section 6.3 proves Theorem 1.3.
where we use that the dynamics depends on the starting configuration σ ∈ S n [m n ] only, through its level magnetisations m n (σ) = m n (see (2.6)), and also use the lumpability.
where τ is the hitting time of the process projected on Γ n .Given the non-degeneracy hypothesis (Hypothesis 1 in Section 1.3.4) and the one-dimensional landscape analysis (in Section 3.3), we can apply [6,Theorem 8.45] to the right-hand side of (6.3) and conclude the proof.
6.3.Randomness of the exponent.In this section we prove Theorem 1.3.In particular, we compute ] to leading order.Recalling definitions (2.26) and (3.5), we have ] k be the critical points of F β,h closest to m n , t n (i.e., the critical points of F n defined above), respectively.Note that m and t satisfy (3.4), while m n and t n satisfy (3.3).Using (2.21), we get and By (3.2), we have (6.7) Thus, Moreover, and Similar equalities hold after we replace t by m and t n by m n .Using the previous computations, we obtain (6.11) Using(6.8),we find we focus on estimating t ,n − t .
From Taylor expansion, we get Since ω ,n t ,n − ω t = (ω ,n − ω ) t + ω ,n (t ,n − t ), (6.15) we have a ω Y t   + o(n −1 ) (6.17) and so 2 ), (6.18)where the denominator does not vanish because of Remark 4.2.Thus, up to a factor O(n − 1 2 ), Y t is a normal random variable with mean 0 and variance (6.19) Similar results hold after we replace t by m.
Going back to (6.12), using (6.13) and (6.18), and inserting t ,n −t ∼ (6.20) Thus, (6.21) Since the random variables Y t , Y m , Z are centred normal, this concludes the proof of Theorem 1.3.
From (6.21) it is possible to compute explicitly the variance of Z defined in Theorem 1.3, because the variances of all the random variables involved are known (at least to leading order).where E denotes expectation with respect to P and U = −β[J(1) K(t) + h], with t solving (3.4).Note that (D.2) is similar to [6,Eq. (14.4.14)].
We are left to find the limit of the determinants ratio.By (4.5),Using the law of large numbers as above and with the same notation, we find 1 − (m ) 2  1 − (t ) 2 , (D.5) where U (x) = −β(J(1)K(x) + h).

2 .
Approximate dynamics and Dirichlet form.For any two vectors v, w ∈ R k , let v, w denote their scalar product.For any k × k matrix M and any v ∈ R k , let M • v denote their matrix product, as v was in R k × 1.

4. 3 .
Approximate harmonic function.Let B n be the k × k matrix defined by(4.18)

( 4 .
29) where we use [6, Eq. (10.2.33)] with ε = 1 βn and d = k.Here 1 2 |A ,n | is the inverse of the step in the -direction, while in [6, Eq. (10.2.33)] the step is ε.Remark 4.3.Note that (4.30) ẼDn (g, g) = ẼDn (g, g) [1 + o(1)] because g(m) = g(m) [1 + o( a unitary flow from ∂ B G n to B. This choice implies that the sum over γ in (5.7) is relevant only on the paths enteringG n in ∂ A G n , exiting G n in ∂ B G n ,and afterwards reaching B without going back to G n .For this purpose we choose f A and f B such that f A ((x, y)) and f B ((x, y)) are proportional to Q n (x).For m ∈ G n such that m ,+ ∈ G n , define(5.8)f ((m, m ,+ )) = Qn (m)r g(m ,+ ) − g(m) + N (g) ,

6. 1 .
Average crossover time.Let us return to the notation of Theorem 1.1, where m n ∈ M n and M n (m n ) = {m ∈ M n \m n : F n (m) ≤ F n (m n )}.To prove Theorem 1.1 we use the relation (6.1) E Γ mn (τ Mn(mn) ) = [1 + o n (1)]

n
and m ,n −m ∼ Y m √ n and ω ,n − ω ∼ Z √ n , we obtain

1 −
(m ) 2 1 + O(n −1 ) .(D.3) Using (3.3) for m ∈ {t n , m n }, we have ∈[k] β a 2 ω ,n [1 − (m ,n ) a local minimum of F n and the correspondent saddle point, respectively, as defined in Section 1.3.4.Note that both m n and t n satisfy (3.3).Consider the neighbourhood of t n defined by Computation of the approximate Dirichlet form.In this section we follow [6, Sections 10.2.2-10.2.3] to approximate ẼDn (g, g) defined in (4.8).As in [6, Eq. (10.2.24)], for m ∈ D n and ∈ 2.33)].The relevant contribution toQ n (A(m n )) is given by the measure of a ball B ρ of radius ρ = C n −1/2 log 1/2 n centred in m n , with C constant, contained in A(m n ).Indeed, if y ∈ A(m n )and d(m n , y) > ρ, then by Taylor expansion of F n around m n we have [6, follows from[6, Theorem 8.15]after proving that M n is a set of metastable points in the sense of[6, Definition 8.2].The latter follows along the lines of the proof of[6, Theorem  10.6], where similar values of capacities and invariant measures occur.Using (6.1) in combination with Proposition 5.1 and Lemma 5.2, we obtain that, for all σ