Metastability for the dilute Curie-Weiss model with Glauber dynamics

We analyse the metastable behaviour of the dilute Curie-Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are Bernoulli random variables with mean $p\in (0,1)$. This model can be also viewed as an Ising model on the Erd\H{o}s-R\'enyi random graph with edge probability $p$. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature $\beta$. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where $N\to\infty$, $\beta>\beta_c=1$ and $h$ is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie-Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities.


INTRODUCTION AND MAIN RESULTS
The randomly dilute Curie-Weiss model (RDCW) is a classical model of a disordered ferromagnet and was studied, e.g. in Bovier and Gayrard in [6]. It generalises the standard Curie-Weiss model (CW) in that the fixed interactions between each pair of spins is replaced by independent, identically distributed, random ferromagnetic couplings between any pair of spins. In [6] it is proven that the RDCW free energy converges, in the thermodynamic limit, to that of the CW model, under some assumptions on the coupling distribution. Their result relies on the fact that the RDCW Hamiltonian can be approximated by that of the CW model up to a small perturbation which can be uniformly bounded in high probability. In the last decade the RDCW model have gained again some attention and various results at equilibrium have been proven, both in the annealed and quenched case. De Sanctis and Guerra [9] give an exact expression of the free energy first in the high temperature and low connectivity regime, and then at zero temperature. The control of the fluctuations of the magnetisation in the high temperature limit is addressed by De Sanctis [8], while recently Kabluchko, Löwe and Schubert [15] prove a quenched Central Limit Theorem for the magnetisation in the high temperature regime.
One of the features which make these random systems with "bond disorder" very appealing is their deep connection with the theory of random graphs, which attracted great interest in the last years due to their application to real-world networks. Indeed, if the random couplings are chosen as i.i.d. Bernoulli random variables with mean p, one can view the model as a spin system on an Erdős-Rényi random graph with edge probability p. There has been an extensive study of the Ising model on different kinds of random graphs, e.g. in Dembo, Montanari [10] and Dommers, Giardinà, van der Hofstad [14], where several thermodynamic quantities where analysed. We refer to van der Hofstad [17] for a general overview of these results.
In contrast to the substantial body of literature on the equilibrium properties of the RDCW model, much less is known about its dynamical properties. The present paper focuses on the phenomenon of metastability for the RDCW model where, for simplicity, the couplings are Bernoulli distributed with fixed parameter p ∈ (0, 1), independent of the number of vertices N, and the system evolves according to a Glauber dynamics. In particular, we give a precise estimate of the mean transition time from a certain probability distribution on the metastable state (called the last-exit biased distribution) to the stable state, when the external magnetic field is small enough and positive and when N tends to infinity. We obtain asymptotic bounds on the probability of the event that the average time is close to the CW one times some constants of order 1 which depend on the parameters of the system.
In the context of metastability for interacting particle systems on random graphs, progress has been made for the case of the random regular graph, analysed by Dommers [12] and for the configuration model, studied by Dommers, den Hollander, Jovanovski, and Nardi [13], both subject to Glauber dynamics, in the limit as the temperature tends to zero and the number of vertices is fixed. In [11] den Hollander and Jovanovski investigate the same model considered in the present paper and obtain estimates on the average crossover time for fixed temperature in the thermodynamic limit. They show that, with high probability, the exponential term is the same as in the CW model, while the multiplicative term is polynomial in N. Their analysis relies on coupling arguments and uses the pathwise approach to metastability.
In contrast, in the present paper, we use the potential theoretic approach initiated by Bovier, Eckhoff, Gayrard and Klein in a series of papers [3,4,5]. (see the monograph of Bovier and den Hollander [2] for an in-depth review). This method has also successfully applied to the random field CW model, where the external magnetic field is given by i.i.d. random variables, by Bianchi, Bovier and Ioffe in [1]. Furthermore, inspired by the results of Bovier and Gayrard [6], namely that the equilibrium properties of the RDCW model are very close to those of the CW model, we observe that, using Talagrand's concentration inequality, the mesoscopic measure can be expressed in terms of that of CW.
Before stating our results we give a precise definition of the model.
where h ∈ R represents an external constant magnetic field, while J i j /N p is a ferromagnetic random coupling. In particular, {J i j } i, j∈ [N] is a sequence of i.i.d. random variables with J i j ∼ Ber(p) and J i j = J ji . The RDCW model can be seen as the Ising model on the Erdős-Rényi random graph with vertex set [N], edge set E and edge probability p ∈ (0, 1) (see van der Hofstad [16] for a general overview on random graphs). In this picture the Hamiltonian can also be written as The Gibbs measure associated to the random Hamiltonian H N is where β ∈ (0, ∞) is the inverse temperature and the partition function is defined as The Gibbs measure µ β,N is the unique invariant (and reversible) measure for the (discrete time) Glauber dynamics on S N with Metropolis transition probabilities where σ ∼ σ ′ means ||σ − σ ′ || = 2 with || · || the ℓ 1 -norm on S N . We denote this Markov chain by {σ(t)} t≥0 the Markov chain and write P ν for the law of the process σ(t) with initial distribution ν conditioned on the realisation of the random couplings. Analogously, E ν is the quenched expectation w.r.t. the Markov chain with initial distribution ν. Moreover, we set P σ = P δ σ . For any subset A ⊂ S N we define the hitting time of A as The Curie-Weiss model. Before stating the main results, we recall some results for the mean-field Curie-Weiss (CW) model (see Bovier and den Hollander [2,Section 13]). The CW Hamiltonian can be obtained taking the mean value of (1.1). An simplifying feature of the CW model is that its Hamiltonian depends on σ only through the empirical magnetisation m N : From now on we will drop the dependency on N from the magnetisation. Theñ and the associated Gibbs measure is whereZ β,N is the normalising partition function. We denote the law of m(σ) under the Gibbs measure byQ is the finite volume free energy, while the entropy of the system is given by the following combinatorial coefficient As N → ∞, (1.14) More precisely, (1.15) We use the notation f β (m) = lim N→∞ f β,N (m). We consider the Glauber dynamics associated to the CW Hamiltonian in analogy with (1.5) and with transition probabilitiesp N (σ, σ ′ ). A particular feature of this model is that the image process m(t) ≡ m(σ(t)) of the Markov process σ(t) under the map m is again a Markov process on Γ N , with transition probabilities (1. 16) The equilibrium CW model displays a phase transition. Namely, there is a critical value of the inverse temperature β c = 1 such that, in the regime β > β c , h > 0 and small, the free energy f β (m) is a double-well function with local minimisers m − , m + and saddle point m * . They are the solutions of equation m = tanh(β(m + h)). Since f β (m − ) > f β (m + ), the phase with m − represents the metastable state, while m + represents the stable state for the system. Defining m − (N), m * (N), m + (N) as the closest points in Γ N to m − , m * , m + respectively, then (m − (N), m + (N)) form a metastable set in the sense of Definition 8.2 of Bovier and den Hollander [2]. Let E CW m − (N) be the expectation w.r.t. the Markov process m(t) with transition probabilitiesr N and starting at m − (N). Then the following theorem holds. . (1.17) We conclude this section by giving the explicit formula of the capacity for the CW model. The definition of capacity is given in (1.27), while its relation with the mean hitting time is given by the key relation (1.26). Let us denote, for any subset U of Γ N , the set of configurations with magnetisation in U by (1.18) and for simplicity, for any m ∈ Γ N , the set of configurations with given magnetisation m by S N [m]. Then, the following formula, (1.20) Since we are going to use ν A,B on the sets S N [m − (N)], S N [m + (N)] defined above, we introduce the following simplified notation (1.21) The following theorem gives a description of the dynamical properties of the RDCW model in the metastable regime where h is positive and small enough, β > β c = 1 (β c is the critical inverse temperature for the CW model) and N is going to infinity. We provide an estimate on the mean time it takes to the system, starting with initial distribution ν N m − ,m + , to reach S N [m + (N)]. More precisely, we estimate, in the limit as N → ∞, its ratio with the mean metastable exit time for the CW model to go from m − (N) to m + (N), providing constant upper and lower bounds independent of N. Because of the random interaction, the result is given in the form of tail bounds.
We are now ready to formulate our main theorem.
Theorem 1.2 (Mean metastable exit time). For β > 1, h > 0 small enough and for s > 0, there exist absolute constants k 1 , k 2 > 0 and C 1 (p, β) < C 2 (p, β, h) independent of N, such that The quantities C 1 and C 2 in the previous theorem can be explicitly written. Set where c 1 , c 2 > 0 are absolute constants coming from Theorem 2.7. It is easy to see that κ < α. With this notation (1.25) 1.4. Proof of the main theorem. The proof of Theorem 1.2 is based on the potential theoretic approach to metastability, which turns out to be a rather powerful tool to analyse the main object we are interested in, i.e. the mean hitting time of S N [m + (N)] for the system with initial distribution ν N m − ,m + . The general ideas of this approach were first introduced in a series of papers by Bovier, Eckhoff, Gayrard and Klein [3,4,5]. We refer to Bovier and den Hollander [2] for an overview on this method.
The crucial formula in the study of metastability is given by the following relation linking mean hitting time and capacity of two sets A, B ∈ S N , The function h AB is called harmonic function and has the following probabilistic interpretation  Theorem 1.4. For any m 1 < m 2 ∈ Γ N and any s > 0, there exist absolute constants k 1 , k 2 > 0 such that asymptotically as N → ∞, where α is defined in (1.23).
We state asymptotic upper and lower bounds on the sum over the harmonic function in the numerator of (1.26) in the following proposition. We used the simplified notation (1.31) Theorem 1.5. For any s > 0, there exist absolute constants k 1 , k 2 > 0 such that asymptotically as N → ∞, and where α and κ are defined in (1.23).
We conclude this section using Theorems 1.3-1.5, to prove the main theorem. First, we introduce the following notation which will be extensively used: for all s > 0 and for some absolute constants k 1 , k 2 > 0, whose values might change along the paper.
Proof of Theorem 1.2 . We prove here only the upper bound, as the lower bound follows similarly. More precisely, we prove We start from (1.26), which in our case reads . . (1.37) Via the lower bound on the capacity from Theorem 1.4, we obtain where we used (1.19) and Theorem 1.1.

1.5.
Outline. The remainder of this paper is organised as follows. In Section 2 we use the powerful Talagrand's concentration inequality to obtain bounds on the equilibrium measure of the RDCW model. These bounds allow us to write the RDCW mesoscopic measure in terms of the deterministic CW one, times a random factor which is the exponential of a sub-Gaussian random variable. In Section 3 we give the proof of Theorems 1.3 and 1.4 via two dual variational principles, the Dirichlet and the Thomson principles, which are the building blocks of the potential theoretic approach to metastability. In obtaining upper and lower bounds on the capacity, the main strategy is to use the results of Section 2 in order to recover the capacity of the CW model. In Section 4 we prove Theorem 1.5, i.e. we compute the asymptotics of the numerator in the formula for the mean hitting time using estimates on the harmonic function.

EQUILIBRIUM ANALYSIS VIA TALAGRAND'S CONCENTRATION INEQUALITY
In this section we prove that the equilibrium mesoscopic measure of the RDCW model is in fact very close to that of the CW model. This is done in two steps. First, we prove that the difference between the random free energy at fixed magnetisation and its average can be controlled via Talagrand's concentration inequality. Second, we find upper and lower bounds on the aforementioned average by estimating first and second moments of the partition function of the RDCW model at fixed magnetisation.
2.1. Mesoscopic measure and closeness to the CW model. We start by analysing the equilibrium measure of the RDCW model. The aim is to express the equilibrium measure µ β,N , defined in (1.3), in terms of the empirical magnetisation in order to obtain a mesoscopic description, as we did for the CW model in Section 1.2. Let us define the measure Q β,N on Γ N , and let the partition function be its normalisation A priori the Hamiltonian of the RDCW model is not only depending on m, but it depends of course on the whole spin configuration. Nonetheless, we will see later in this section that the mesoscopic measure Q β,N can be written in terms of the mesoscopic measureQ β,N of the standard CW model. We first notice that the expectation of the RDCW Hamiltonian is the CW one, i.e.
Therefore, we can split the Hamiltonian into the mean-field part and the remaining random part obtaining Note that ∆ N,p is a random variable with zero mean. In order to simplify the notation, we drop from now on the dependence on N and p, from ∆ N,p . Next, we write the mesoscopic measure as where E(m) is defined in (1.8). We introduce the following notation, where we drop the dependence on β for simplicity where Z N,m can be interpreted as a partition function for a system of spins with fixed magnetisation, F N,m as the associated random free energy and p N,m as its average. We are interested in finding precise estimates on Z N,m by writing it in terms of the entropic exponential term e −NI N (m) times some random factor which takes into account the randomness of the couplings. We notice that Z N,m is the product of a deterministic factor e N p N,m and a random factor e N(F N,m −p N,m ) .
We first characterise the random variable N(F N,m − p N,m ) in the following Proposition.
Proposition 2.1. For any β, any t > 0, where γ ∝ p 2 β 2 . The previous result intuitively means that the random free energy F N,m is in fact very well concentrated around its mean p N,m .
As a second step we provide asymptotic bounds on the average of F N,m , i.e. the deterministic term p N,m .
where κ is defined in (1.23).  12) and where c is a positive constant.
We prove Proposition 2.1 in Section 2.2, and the Lemmas 2.2 and 2.3 in Section 2.3. We proceed now by giving the main result of this section, as a corollary of Proposition 2.4.

(2.17)
The lower bound is proven similarly.
We conclude this section by introducing some notation which will be widely used later.
Property 2.6. Let Y be a sub-Gaussian random variable such that where k 1 , k 2 > 0 are absolute constants, and consider the random variable X = exp(Y). For all s > 0, it is trivial to see that Sub-Gaussian bounds on the random term. Proposition 2.1 follows from Talagrand's concentration inequality, which we cite for completeness in the version of Tao [19].
Then, for any t ≥ 0, concluding the proof of (2.9) and hence Proposition 2.1.

Asymptotic bounds on
In order to find estimates for (2.22), we first define which is a function independent of i, j, being {Ĵ i j } i, j i.i.d., with first and second derivatives Performing a Taylor expansion of Φ we get where we used the expansion log(1 + x) = x + o(x). Therefore, for any sequence of coefficients x 2 i j which are independent of i, j and σ, we have the following asymptotically, for x i j → 0, where the second equality holds only if x 2 i j is independent of i, j and the last equality holds only if x 2 i j is independent of σ. Applying (2.28) to Therefore, by Jensen's inequality and (2.29), we have which proves the upper bound.

Proof of Lemma 2.3.
A key ingredient in the proof is to control the upper bound on the second moment of Z N,m , i.e. prove that the following bound holds where α is defined in (1.23).
where in the second line we used again (2.28) with We recall the Paley-Zygmund inequality, which states that for any non negative random variable X and any η ∈ (0, 1). By (2.29), (2.32) and (2.33) we get, asymptotically as N → ∞, .
(2.34) Moreover, after a change of variables in (2.21), we obtain ∀ t > 0, Next we prove that the intersection of the events in (2.34) and (2.36) is non empty. Assuming, for η ∈ (0, 1), that and comparing (2.34) and (2.37), we notice that the sum of the probabilities of the two is strictly greater than 1. Therefore, they intersect in the not empty event As a consequence, the latter set is non empty and, being deterministic, holds with probability 1. It remains to choose a suitable t > 0 for assumption (2.37) to hold. A sufficient condition is, for every η ∈ (0, 1), (2.44) Therefore, we obtain, for every η ∈ (0, 1), Notice that κ η < α. In order to obtain the best lower bound, namely the closer to the upper bound proven in Lemma 2.2, we choose η ∈ (0, 1) s.t. α−κ η is minimised and we conclude the proof. This choice motivates the maximum in the definition of κ, in (1.23).

CAPACITY ESTIMATES
This section in entirely devoted to obtain upper and lower bounds on capacities between sets with a fixed magnetisation. These bounds are obtained via two dual variational principles, i.e. the Dirichlet and Thomson principles which are extensively discussed in Bovier and den Hollander [2]. The result will be expressed in terms of the capacity for the Curie-Weiss model, see (1.19). In particular, we prove Theorem 1.3 in Section 3. where (3.2) Later it will be clear that we can restrict the previous variational principle over the functions on the space Γ N , hence it is useful to definẽ In order to simplify the notation we will often neglect the dependency on m 1 , m 2 when this will not generate confusion. From (3.1), we have We turn now to the last sum in (3.4) and call this quantity G(σ, m ′ ). If σ ∼ σ ′ , then σ and σ ′ differ on a single state, say ℓ ∈

5)
where in (3.8) we have used the following elementary facts holding asymptotically in N,  (1)) . ≤ e s+2β+αZ β,N min where we used the notation (1.34). Furthermore, we noticed that the variational form appearing in the previous inequality is given by the Dirichlet principle applied to the CW model and therefore it is equal to the capacity of the CW model. . For all σ, σ ′ ∈ S N , we define the candidate flow Ψ N as follows (3.14) where, for all m ∈ Γ N , The proof of Theorem 1.4 is postponed after three technical intermediate results which are essential for it. The following lemma allows us to use Ψ N in the Thomson principle.  After a similar computation for the right hand side, we obtain that (3.19) is satisfied.

Lemma 3.2. For all σ ∈ S N , the following holds
where we have used that, since h > 0, In the proof of Theorem 1.4 we will need an upper bound on σ∈S N [m] exp (β∆(σ)). Noticing the analogy with (2.6) one proves the following Lemma, which is very similar to Proposition 2.4. We introduce the following overlined notation, in analogy to (2.6)-(2.8), With the previous lemmas at hand, we are now ready to prove the lower bound on the capacity.

Proof of Theorem 1.4. By the Thomson principle (see Bovier and den Hollander [2, Section 7.3.2]) we have
, (3.27) where Ψ N is the test flow we defined in (3.14), which by Lemma 3.1 is in Thus, we are interested in upper bounds on . (3.28)

By multiplying and dividing by exp(−β[H
(3.29) We use Lemma 3.3 to bound the sum over σ and Lemma 3.2 to bound the sum over σ ′ , obtaining Substituting into (3.28) the flow Ψ N defined in (3.14) -(3.15) and using Property 2.6, we obtain where we used the notation (1.34). Therefore, by (3.27) and (3.31), we obtain where we used the notation (1.34) and we noticed that the inverse of the expression appearing in brackets in (3.32) gives exactly the capacity for the CW model. Indeed, to compute the capacity for the CW model with Glauber dynamics, one can simply notice that it is equivalent to a one-dimensional random walk in Γ N and use the formula for the capacity in Bovier and den Hollander [2, Section 7.1.4].

ESTIMATES ON THE HARMONIC FUNCTION
As pointed out in Section 1.4, the proof of Theorem 1.2 relies on sharp estimates on capacities, carried out in Section 3, and estimates on the harmonic function. We entirely devote this section to obtain asymptotic upper and lower bounds on the numerator in (1.36), which is given by the following sum that is to give the proof of Theorem 1.5. In order to control the sum (4.1), one generally uses a renewal argument which relies again on estimates over capacities. However, in our case this is not possible, due to the fact that capacities of single spins are too small.
We first prove the upper bound and then give some details about how to prove the lower bound, which is very similar and more straightforward. Our proof follows Bianchi, Bovier and Ioffe [1, Section 6].

4.1.
Notation and decomposition of the space. Before starting with the proof, we introduce some notation. We refer to Figure 1 below for a better visual understanding of the objects we are defining.
Recall that we denote by m + the global minimum, by m − the local minimum, and by m * the local maximum of f β (·) in [−1, 1], where f β (·) = lim N→∞ f β,N (·), defined in (1.12). We want to decompose the space Γ N (and eventually the set of spin configurations S N ) according to the values of f β . The notation and the decomposition is organised in 4 steps.
Step 1. First, let δ > 0 be small in a way which will become clear later, and define the set \ U δ and we denote by U δ (m) the connected component of U δ containing m. Note that {m − , m + } ∈ U δ . In general, U δ (m − ) and U δ (m + ) may have non empty intersection, but we choose δ such that m * U δ , implying that U δ is partitioned by the disjoint sets U δ (m − ) and U δ (m + ). For this to hold, it suffices to take δ < f β (m * )− f β (m − ). Furthermore, let us denote by m δ the unique point in (m * , m + ) such that Step 2. With δ chosen as above, we define a sequence (δ N ) N∈N , converging to δ from below, such that the left extreme of U δ N (m + ) is in Γ N . Specifically, we define δ N as follows: 4) for N sufficiently large. Moreover, set for all m ∈ [−1, 1]. Thus, we have the partition Remark. Notice that, for N sufficiently large, U δ,N (m − (N)) = U δ,N (m − ) and U δ,N (m + (N)) = U δ,N (m + ). Furthermore, with these definitions, m δ N ∈ U δ,N and it is the left extreme of U δ,N (m + ).
Step 3. Let ε > 0 be arbitrarily small (the choice of ε will be relevant in Section 4.2). We denote by m ε the only point in a small left neighbourhood of m + , more precisely in Step 4. Similarly to Step 2, fixed ε > 0, we want to define a sequence (ε N ) N∈N converging to ε from below such that m ε N is in Γ N . More precisely, we define ε N as follows (4.9) We will use later that m ε N ∈ U δ,N (m + ) and it satisfies f β (m ε N ) = f β (m + ) + ε N . Moreover, given ε > 0, we define the sequence (θ N ) N∈N , analogously to (4.8), by set-

4.2.
Upper bound on the harmonic sum. In this section we prove the first part of Theorem 1.5 by giving an upper bound on the harmonic sum in (4.1).
With the notation introduced in Steps 1-4 in Section 4.1, we partition S N as follows The proof of the latter result is quite technical and it is postponed to Section 4.3. We now state Lemma 4.1, which allows us to find bounds on the random mesoscopic measure Q β,N in terms of the infinite volume CW free energy f β . For all m ∈ (−1, 1), for all s > 0, and, for m ∈ {1, −1}, where we used the notation (1.34).
The proof of Lemma 4.1 is straightforward and uses Corollary 2.5, Property 2.6 and the following useful expansion. . (4.14) In the last step we first approximated, for N sufficiently large, the sum with an integral and then applied the saddle point method (see, for instance de Bruijn [7, Chp 5.7]), where m − is the maximum point of −β f β on the considered domain. More precisely, where the last inequality holds by definition of U c δ,N .
(4.19) Thus, applying Lemma 4.3 below, we have (1)). (4.20) From the bound in Lemma 4.1, we obtain . (4.21) Now we prove that this part is not relevant compared to the right hand side of (4.14). In particular, we show that, for a certain choice of γ, is positive and its limit, ) is positive and finite. In order to achieve this, we choose γ ∈ (0, 1) small enough, such that c N and its limit are positive, definitely in N. In particular, we want to impose definitely in N, and First, we notice that it is easy to check that the previous quantities are strictly smaller than 1. Second, we want to show that a strictly positive γ satisfying (4.24)-(4.25) exists. Note that ℓ N (θ N ), defined in (4.39), has the following trivial upper bound for every N, (4.26) Thus, a sufficient condition is to choose, for N large enough, γ ≥ γ 0 , where is clearly strictly positive. Indeed, we can choose ε > 0 sufficiently small for the numerator on the left hand side of (4.27) to be positive, while θ is small accordingly to ε. We conclude by obtaining, for N sufficiently large,

Conclusion.
With the previous bounds at hand, we are now ready to conclude the proof of the upper bound. Decomposing the sum over S N using (4.10), and inserting the estimates we computed above into (4.1), we obtain The latter relies on Lemma 4.6, which we subsequently prove using Lemma 4.7. We conclude the section proving Lemma 4.7. Throughout this section we will use the notation introduced in Section 4.1.  N)) , for all γ ∈ (0, 1) and ε > 0,
Proof. For all σ ∈ S N [m δ N , m + (N)) , we have where we notice that, Using the Markov property and taking the maximum of the first factor out of the sum, we have that, for all σ ∈ S N [m δ N , m + (N)) , (4.33) We first consider the case σ ∈ S N [m ε N ]. By Lemma 4.4, we get (1)) .

(4.34)
Taking the maximum over σ and noticing that the same term appears in both right and left hand side of the inequality, we obtain, for all where we used Lemma 4.5.
By Taylor expansion of f β m ε N − 2 N and definition of m ε N , we get max (1)), (4.36) where the last inequality holds for N sufficiently large.
Now we consider the case where σ ∈ S N [m δ N , m + (N)) \ {m ε N } . Going back to (4.33) and using again (4.36) we obtain In the last inequality we used Lemma 4.6, which holds for Remark. In Lemma 4.3 one might try to further bound the r.h.s. of (4.30) using that This would yield to the trivial upper bound 1 on P σ (τ S N [m − (N)] < τ S N [m + (N)] ), which is not sufficient for our purpose of proving that the second term in (4.29) is negligible with respect to the last one. The way to go is, therefore, to keep the dependence on m(σ) in order to obtain later a more suitable bound, uniform in m, by exploiting the smallness of Q β,N (m(σ)) in (4.20) and (4.21).
In order for (4.34) to be true, we have to prove the following result.  (4.38) where ℓ N : R → R is defined by We want to partition A N (σ) according to the values of the first k+1 elements of its paths. Given a sequence π ∈ S k+1 N , let us denote by {π} the set of all paths in A N (σ) in which the first k + 1 elements are exactly given by π, namely {π} = {(σ(0), σ(1), . . . , σ(k), σ(k + 1), . . . ) ∈ A N (σ) : (σ(0), . . . , σ(k)) = π} . (4.41) Notice that, by definition of A N (σ), {π} is empty for many π ∈ S k+1 N . We denote by B N (σ) the set of all the sequences π ∈ S k+1 N such that {π} is not empty. Thus, we obtain the following partition of A N (σ) {π}. (4.42) Fix σ ∈ S N [m ε N ], then one simply notices that Thus, we first find a lower bound on P σ ({π}) independent of π in B N (σ) and later we compute the cardinality of B N (σ). Fix π = (σ(0), σ(1), σ(2), . . . , σ(k)) ∈ B N (σ), then we have where m i = m(σ(i)), C = exp (−β|2 − 2h|) and we used the following fact where r is the index of the spin to be flipped to go from σ(i−1) to σ(i). Therefore, recalling that m i ∈ [m ε N , m + (N)], we obtain the following lower bound independent of π Indeed, for ε N sufficiently small, m ε N is close to m + (N) > 0, allowing us to assume m ε N > 0. Therefore, −2m ε N − 2 N − 2h < 0, which implies the last equality in (4.46). We are left to compute the cardinality of B N (σ), with σ ∈ S N [m ε N ], namely we have to count all paths from σ to S N [m + (N)] with increasing magnetisation and length k + 1. Any of these paths is characterised by a final spinσ ∈ S N [m + (N)] and a sequence of negative spins which are flipped. Notice thatσ is reachable by σ through a path with increasing magnetisation if and only if the two following properties are satisfied:σ has k positive spins more than σ and, for all i ∈ [N], σ i = +1 impliesσ i = +1. Thus, a configurationσ ∈ S N [m + (N)] reachable by σ through a path with increasing magnetisation is characterised by the k spins which are negative in σ and positive inσ. Therefore, the number of reachable configurationsσ is The number of paths with increasing magnetisation from σ ∈ S N [m ε N ] to a reachablē σ ∈ S N [m + (N)], both fixed, is k!, namely the number of permutations of the k negative spins which are flipped along a path. Thus, being k = 1 2 Nθ N , the cardinality of B N (σ) is Going back to (4.43), we obtain (4.49) Using Stirling's approximation n! = √ 2πn n n e −n (1 + o(1)) = √ 2πn e n(log n−1) (1 + o(1)) and the notation we obtain (1)). (4.51) Thus, since k θ N ≥ 1 and C = exp(−β|2 − 2h|), we conclude by (4.52) To prove Lemma 4.3 we used the following fact.
Lemma 4.5. For σ ∈ S N [m ε N ], for N sufficiently large and any γ ∈ (0, 1), (4.54) The first term vanishes because all the probabilities in the sum are zero. Thus, we get the upper bound Using first Lemma 4.6 and then Lemma 4.2, we obtain In the proofs of Lemmas 4.3 and 4.5, we use the following fact.
For a suitably chosen ψ the latter inequality will yield the desired upper bound. Now we are left with the choice of a suitable ψ : S N → R such that Lψ(x) < 0, for all x ∈ S N [m δ N , m ε N ) . We define a function ψ which depends on a parameter γ ∈ (0, 1) and is constant on fixed magnetisation sets, i.e, for all σ ∈ S N , ψ(σ) = φ(m(σ)),  where we used the definition of m δ N .
We are now left with the proof of the super-harmonicity of ψ, which is used in the proof of Lemma 4.6. Proof. We have to prove that Lψ(x) < 0, for all x ∈ S N [m δ N , m ε N ) . Fix x in S N [m δ N , m ε N ) and use the notationm = m(x). As usual, we try to rewrite the terms appearing in the expression for Lψ(x) in terms of their mean-field version.  Then, recalling that I ′ (m) = 1 2 log 1+m 1−m , and using (4.70) we have (4.72) Therefore, rearranging (4.69) and using (4.71) and (4.72), we obtain

4.4.
Lower bound on the harmonic sum. In this section we provide the main ideas to prove the second part of Theorem 1.5, namely the lower bound on the harmonic sum in (4.1).
Proof of Theorem 1.5. Lower bound. The proof is very similar to the proof of the upper bound we gave in Section 4.2, therefore we omit the details. The main contribution is given once again by the sum on S N U δ,N (m − ) .
We have, (4.74) The first term, i.e. the sum on the mesoscopic measure Q β,N , gives the main contribution. This sum can be estimated from below by using the second bound in Corollary 2.5 and then obtaining a lower bound similar to the one in Lemma 4.1. More precisely, using the notation (1.34), we have the following lower bound for s > 0 :