{'fraction_non_alphanumeric': 0.06394661999562459, 'fraction_numerical': 0.02288339531831109, 'mean_word_length': 3.41780226152508, 'pattern_counts': {'":': 0, '<': 31, '<?xml version=': 0, '>': 15, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 39, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Sum of exit times in a series of two metastable states

We consider the problem of non degenerate in energy metastable states forming a series in the framework of reversible finite state space Markov chains. We assume that starting from the state at higher energy the system necessarily visits the second one before reaching the stable state. In this framework, we give a sharp estimate of the exit time from the metastable state at higher energy and, on the proper exponential time scale, we prove an addition rule. As an application of the theory, we study the Blume-Capel model in the zero chemical potential case.


Introduction
Metastable states in finite volume Statistical Mechanics lattice systems, in which stochastic transition between states are controlled by an energy function, is a well understood phenomenon. Different mathematical theories have been developed in the last decades. The pioneering pathwise approach [21,24,25], further developed for non-reversible dynamics in [15], and the more recent potential theoretic approach [5,6,26], further developed in [1] via the use of the trace process, allow a thorough description of the phenomenon.
The first theory gives a handy definition of the metastable state and a physically clear interpretation of the associated exit time. In the low temperature limit, it has been proven indeed that the time scale on which the system leaves the metastable state is exponentially in [16].
The paper is organized as follows: in Section 2 the general model is introduced and our main results are stated. Those results are then proved in Section 3. Finally, the application to the Blume-Capel model is discussed in Section 4.

Model and results
In this section we first introduce a general reversible Markov chain and specify the conditions on the energy landscape in order to have a series of metastable states. We next state our main results in this framework.

Reversible Markov chains
We want to give the notion of reversible Markov chain [25, Condition R, page 335] in a quite general setup so that the theory will apply to different and relevant examples such as Statistical Mechanics Lattice models and the reversible Probabilistic Cellular Automata.
Consider a finite state space X and a family of irreducible and aperiodic Markov chain x(t), with t ∈ Z + parametrized by the parameter β > 0, called inverse temperature. We let p β (x, y) and µ β (x), for x, y ∈ X be respectively the transition matrix and the stationary measure. We assume that the Markov chains are reversible with respect to µ β , namely, µ β (x)p β (x, y) = p β (y, x)µ β (y) (2.1) for any x, y ∈ X. We also assume that the fact that a jump is not allowed does not depend on β, namely, if p β (x, y) = 0 then p β (x, y) = 0 for any β .
The definition of the model will be completed by assuming a slightly enforced version of the well known Wentzel-Friedlin condition and by requiring that the stationary measure is "close" to a Gibbs one 1 . More precisely, we assume that there exists ∆ : X × X → R + ∪ {∞} and r : X × X → R such that, for any x, y ∈ X, ∆(x, y) = ∞ if p β (x, y) = 0 and lim β→∞ [− log p β (x, y) − β∆(x, y)] = r(x, y) if p β (x, y) > 0. (2.2) Note that ∆ and r do not depend on the inverse temperature. We shall call ∆ the cost function.
Moreover, we assume that there exist two functions s, H : X → R and a family of functions G β : X → R parametrized by β > 0 such that for x ∈ X and max x∈X s(x) =s < ∞. The normalization factor in µ β is denoted by Z β and called partition function. Note that the function H does not depend on the inverse temperature. We shall call H the Hamiltonian or energy of the model. From (2.2) and (2.3) it follows immediately that lim β→∞ 1 β log p β (x, y) = −∆(x, y) for all x, y ∈ X such that p β (x, y) > 0 and for x ∈ X. From (2.1) and the conditions above it follows also that H(x) + ∆(x, y) = ∆(y, x) + H(y) (2.4) for all x, y ∈ X.
For any x ∈ X, we denote by P x (·) and E x [·] respectively the probability and the average along the trajectories of the process started at x.

Examples
In this section we discuss two important examples of dynamics fitting in the general scheme depicted above. The Metropolis dynamics on the states space X with energy K : X → R, inverse temperature β, and connection matrix q(x, y) is defined by letting It is well known that the Metropolis dynamics has as stationary measure the Gibbs measure with Hamiltonian K, so that the condition (2.3) is satisfied with G β = βK and s = 0. Now, we let r(x, x) = 0 and r(x, y) = − log q(x, y) for x = y and q(x, y) > 0. For any x = y, we let also ∆(x, y) = ∞ if q(x, y) = 0 and ∆(x, y) = [K(y) − K(x)] + otherwise. Finally, we let ∆(x, x) = ∞ if p β (x, x) = 0 and ∆(x, x) = −(1/β) log p β (x, x) otherwise. It is immediate to verify that condition (2.2) is satisfied for any x, y ∈ X.
A second important example is that of reversible Probabilistic Cellular Automata (PCA). Reversible PCA have been introduced in [19], see also [12] for a detailed discussion, and cns-pta2016.tex -14 settembre 2016 provide a very interesting example of dynamics with a parallel updating rule which are reversible with respect to a stationary measure which is very close to a Gibbs measure. The metastable behavior of some reversible Probabilistic Cellular Automata has been firstly studied in [2].
Let Λ ⊂ Z 2 be a finite cube with periodic boundary conditions. Associate with each site i ∈ Λ the state variable x i ∈ {−1, +1} and denote by X = {−1, +1} Λ the state space. For any x ∈ X we consider on {−1, +1} the probability measure f x,β (s) = 1 2 1 + s tanh β j∈Λ k(j)x j + h for s ∈ {−1, +1}, where β > 0 and h ∈ R are called inverse temperature and magnetic field respectively. The function k : Z 2 → R is such that its support is a subset of Λ and k(j) = k(−j) for all j ∈ Λ. Recall that, by definition, the support of the function k is the subset of Λ where the function k is different from zero. We assume, also, that for any x ∈ X and i ∈ Λ. We finally introduce the shift Θ i on the torus, for any i ∈ Λ, defined as the map Θ i : for x, y ∈ X. We remark that the character of the evolution is parallel, in the sense that at each time all the spins are potentially flipped. It is not difficult to prove [19] that the above specified PCA dynamics is reversible with respect to the finite-volume Gibbs-like measure The low-temperature behavior of the stationary measure of the PCA can be guessed by looking at the function The difference between F β and βK can be computed explicitly, indeed in [12] it is proven that for each β > 0 and x ∈ X. From the remarks above, recall also the assumption (2.5), it follows immediately that the reversible PCA satisfies condition (2.3) with G β = F β , H = K, and s(x) = |Λ| log 2 for any x ∈ X.
As for the transition rates, we set Thus, the reversible PCA satisfies condition (2.2) with ∆ = V and r(x, y) = 0 for any x, y ∈ X. For completeness, we finally prove (2.6). In the following computation we shall use many times the assumption (2.5). First note that yielding (2.6).

Energy landscape
After the short "intermezzo" on the Metropolis and the reversible PCA models, we come back to the general setup of Section 2.1. Let Q be the set of pairs (x, y) ∈ X × X such that p β (x, y) > 0 or, equivalently, ∆(x, y) < ∞. The quadruple (X, Q, H, ∆) is then a reversible energy landscape [14].
Given Y ⊂ X we let its external boundary ∂Y be the collection of states z ∈ X \ Y such that there exists y ∈ Y such that (y, z) ∈ Q. In words, the external boundary is made of those states outside Y such that there exists a state in Y where the system can jump.
Given Y ⊂ X such that H(y) = H(y ) for any y, y ∈ Y , we shall denote by H(Y ) the energy of the states in Y . For any Y ⊂ X we shall denote by F (Y ) the set of the minima of the energy inside Y , that is to say y ∈ F (Y ) if and only if H(y ) ≥ H(y) for any y ∈ Y . We let X s := F (X) be the set of ground states of H, namely, the set of the absolute minima of the energy.
For any positive integer n, ω ∈ X n such that (ω i , ω i+1 ) ∈ Q for all i = 1, . . . , n − 1 is called a path joining ω 0 to ω n ; we also say that n is the length of the path. For any path ω of length n, we let be the height of the path 2 . For any y, z ∈ X we denote by Ω(y, z) the set of the paths joining y to z. For any y, z ∈ X we define the communication height between y and z as where we have used the notation Ω(Y, Z) for the set of paths joining a state in Y to a state in Z. For any y, z ∈ X we define also the communication cost from y to z as the quantity Φ(y, z) − H(y). Note that in general the communication cost from y to z differs from that from z to y.

Metastable states
For any x ∈ X we denote by I x the set of states y ∈ X such that H(y) < H(x). Note that I x = ∅ if x ∈ X s . We then define the stability level of any x ∈ X \ X s Note that the stability level V x of x is the minimal communication cost that, starting from x, has to be payed in order to reach states at energy lower than H(x).
Following [21] we now introduce the notion of maximal stability level. Assume X \X s = ∅, we let the maximal stability level be Definition 2.1 We call metastable set X m , the set Note that, since the state space is finite, the maximal stability level Γ m is a finite number. Following [21], that is to say by assuming the so called pathwise point of view, we shall call X m the set of metastable states of the system. Each state x ∈ X m is called metastable. A different, even if strictly related, notion of metastable states is that given in [5] in the framework of the Potential Theoretic Approach. First recall that the Dirichlet form associated with the reversible Markov chain is defined as the functional where f : X → R is a generic function. Thus, given two not empty disjoint sets Y, Z ⊂ X the capacity of the pair Y and Z can be defined as cap β (Y, Z) := min Note that the capacity is a symmetric function of the sets Y and Z. It can be proven that the right hand side of (2.15) has a unique minimizer called equilibrium potential of the pair Y and Z and denoted by h Y,Z . A nice interpretation of the equilibrium potential in terms of hitting times can be given. For x ∈ X and Y ⊂ X we shall denote by τ x Y the first hitting time to Y of the chain started at x. Whenever possible we shall drop the superscript denoting the starting point from the notation. Then, it can be proven that where τ Y and τ Z are, respectively, the first hitting time to Y and Z for the chain started at x. It can be proven that, for any Y ⊂ X and z ∈ X \ Y , see [4, equation (3.10)].
The prefix p.t.a. stands for potential theoretic approach. We used this expression in order to avoid confusion with the set of metastable states X m introduced in (2.13). The physical meaning of the above definition can be understood once one remarks that the quantity µ β (x)/cap β (x, y), for any x, y ∈ X, is strictly related to the communication cost between the states x and y, see Proposition 1.1. Thus, condition (2.18) ensures that the communication cost between any state outside M and M itself is smaller than the communication cost between any two states in M . In other words, it states that getting to M starting from any state outside M is "much" easier than going from any point in M to any other point in M . Finally, given a p.t.a.-metastable set M ⊂ X, for any x ∈ M we let be the valley associated with x. For any y ∈ X and any z ∈ M the quantity P y (τ z = τ M ) measures the probability that, starting from y, the system touches M for the first time in z. Thus, by computing sup z,∈M P y (τ z = τ M ), one detects the best way to touch M for the system started at y. Hence, the condition P y (τ x = τ M ) = sup z,∈M P y (τ z = τ M ) selects all the cns-pta2016.tex -14 settembre 2016 9 0:45 sites y such that, for the system started at y, the best way to touch M is that of touching it for the first time at x.

Series of metastable states
The aim of this paper is that of proving an addition formula for the exit time from metastable states in the case in which they form a series. With this expression we mean that the structure of the energy landscape is such that the system has two non degenerate in energy metastable states and the system, started at the one having higher energy, must necessarily pass through the second one before relaxing to the stable state. See Fig. 2.1 for a schematic description of the situation we have in mind and that will be formalized through the following conditions. Condition 2.3 Recall (2.12) and (2.13).We assume that the energy landscape (X, Q, H, ∆) is such that there exist three states x 2 , x 1 , and x 0 such that Note that, by recalling the definition of the set of ground states X s , we immediately have that Moreover, from the definition (2.12) of maximal stability level it follows that (see [14, Theorem 2.3]) the communication cost from x 2 to x 0 is equal to that from x 1 to x 0 , that is to say Note that, since x 2 is a metastable state, its stability level cannot be lower than Γ m . Then, recalling that H(x 2 ) > H(x 1 ), one has that Φ(x 2 , x 1 ) − H(x 2 ) ≥ Γ m . On the other hand, (2.21) implies that there exists a path ω ∈ Ω(x 2 , x 1 ) such that Φ ω = H(x 2 ) + Γ m and, hence, The two bounds finally imply that Note that the communication cost from x 0 to x 2 and that from x 1 to x 2 are larger than Γ m , that is to say, Indeed, by recalling the reversibility property (2.9) we have where in the last two steps we have used (2.22) and Condition 2.3, which proves the second of the two equations (2.23). The first of them can be proved similarly. We want to implement in the model the series structure depicted in Fig. 2.1. With this we mean that when the system is started at x 2 with high probability it will visit x 1 before x 0 . For this reason we shall assume the following condition.

Condition 2.4 Condition 2.3 is satisfied and
We remark that the Condition 2.4 is indeed a condition on the equilibrium potential h x 0 ,x 1 evaluated at x 2 . The most important goal of this paper is that of proving the formula (2.28) for the expectation of the escape time τ x 0 for the chain started at x 2 . Such an expectation, hence, will be of order exp{βΓ m } and the prefactor will be that given in (2.28). At the level of logarithmic equivalence, namely, by renouncing to get sharp estimate, this result can be proven by the methods in [21]. More precisely, one gets that (1/β) log E x 2 [τ x 0 ] tends to Γ m in the large β limit.
We can thus formulate the further assumptions that we shall need in the sequel in order to discuss the problem from the point of view of the Potential Theoretic Approach.

Condition 2.5 Condition 2.3 is satisfied and there exists two positive constants
where o(1) denotes a function tending to zero in the limit β → ∞.

Main results
We shall prove the addition rule for the exit times from the metastable states by using the sharp estimates provided by the Potential Theoretic Approach to metastability originally developed in [5].
By means of the theory in [5] it is possible to write asymptotic estimates of the first hitting time to a subset of a p.t.a.-metastable set when the dynamics is started in state of the same p.t.a.-metastable set not belonging to the considered subset. These results, see for instance [5,Theorem 1.3], are typically proven under suitable not degeneracy conditions [5, Definition 1.2] that are not satisfied in our case, due to the presence of multiple metastable states. In the following theorem we state two results that, for the reasons outlined above, can be deduced directly from those in [5]. On the other hand, as we shall discuss in detail in Section 3, they can be deduced by some of the results proven in [4] (see, also, [6]). But, since we assumed strong hypotheses on the energy landscape of the model, it will be possible to prove the theorem directly by means of simple estimates. This "ad hoc" proof is also given in Section 3.

Theorem 2.7 Assume Conditions 2.3 is satisfied. Then
We remark that Theorem 2.9 gives an addition formula for the mean first hitting time to x 0 starting from x 2 . Neglecting terms of order o(1), such a mean time can be written as the sum of the mean hitting time to the pair {x 1 , x 0 } when the chain is started at x 2 and of the mean hitting time to x 0 when the chain is started at x 1 . It is very interesting to note that no role is plaid in this decomposition by the mean hitting time to x 1 for the chain started at x 2 . Indeed, on the exp{βΓ m } time scale no control can be proven for such a mean time. For instance, in the case of the Blume-Capel model that will be studied in Section 4, in [20,Proposition 2.5] it is proven that E x 2 [τ x 1 ]/e βΓm diverges in the limit β → ∞.

Proof of results
In this section we proof the theorems stated above and related to the general setup given in Section 2.5.
We just note that the Theorem 3.6 in [14] has been proved in a slightly different context, but the proof given there applies also to the more general case studied here.
Theorem 2.7 can be deduced by using the structure provided by Theorem 2.6 above, and the general results [4, Eq. Before discussing such a proof we state two useful lemmas. Recall Condition 2.3, in the first of the two lemmas we collect two bounds to the energy cost that has to be payed to go from any state x = x 1 to x 1 or to x 0 . The second lemma is similar.
Proof. Let us prove the first inequality. By Theorem 2.3 in [14] we have that Φ(x, x 0 ) ≤ Γ m + H(x). If by contradiction Φ(x, x 0 ) = Γ m + H(x) then, by the same Theorem 2.3 in [14], x ∈ X m which is in contradiction with Condition 2.3.
As regards the proof of the second inequality we distinguish two cases. Case H(x) < H(x 1 ): we have that x ∈ I x 1 . By Definition 2.1 of metastable state and by (2.11), we get that proves the inequality.
Case H(x) = H(x 1 ): let us define the set that proves the inequality. the inequality.
Proof. Let us prove the first inequality. By Theorem 2.3 in [14] we have that Φ( then, by the same Theorem 2.3 in [14], x ∈ X m which is in contradiction with Condition 2.3. As regards the proof of the second inequality we distinguish two cases. Case H(x) < H(x 2 ): we have that x ∈ I x 2 . By Definition 2.1 of metastable state and by (2.11), we get that proves the inequality.
Case H(x) = H(x 2 ): let us define the set that proves the inequality.
Proof of Theorem 2.7. We prove in details the right of equation ( Considering the contribution of x 1 in the sum and recalling (2.16), we get the following lower bound: In order to provide un upper bound, we first use the boundary conditions in (2.16) to rewrite (3.3) as follows: Recalling that h x 1 ,x 0 (x 1 ) = 1, the equilibrium potential is not bigger than one, the configuration space is finite, and µ β (x) = µ β (x 1 ) exp{−βδ} for some positive δ and for any By (2.16) and (1.2) we can give the following upper bound for the equilibrium potential where in the first inequality we used Proposition 1.1, in the second Lemma 3.10, and C, δ are suitable positive constants. By using (3.5) we get Which implies cns-pta2016.tex -14 settembre 2016 where we have used that the configuration space is finite. The Theorem finally follows by (3.4) and (3.6).
Proof of Theorem 2.8. The theorem follows immediately by exploiting Condition 2.5 and applying Theorem 2.7.
The proof of Theorem 2.9 is based on the following lemma.
Lemma 3.12 Given three states y, w, z ∈ X pairwise mutually different, we have that the following holds Proof. First of all we note that We now rewrite the first term as follows where we have used the fact that τ w is a stopping time, that I {τw<τz} is measurable with respect to the pre-τ w -σ-algebra F τw and the strong Markov property which gives Proof of Theorem 2.9. By (3.7) we have that By Theorem 2.8 and Condition 2.4 it follows that

Application to the Blume-Capel model
In this section, as a possible application of the theory described above, we apply our results to the case of the Blume-Capel model ( [3,10]). In particular, we consider the model with cns-pta2016.tex -14 settembre 2016 null chemical potential, which has two metastable states non degenerate in energy. We shall then derive, in a different way, the results already appeared in [20]. Let us consider a square lattice Λ ⊂ Z 2 with periodic boundary conditions and side length L. Let {−1, 0, +1} be the single spin state space and X := {−1, 0, +1} Λ be the configuration space. The Hamiltonian of the model [14] is for any σ ∈ X , where the first sum runs over the pairs of nearest neighbors and h ∈ R is the magnetic field. We denote by µ β the corresponding Gibbs measure with inverse temperature β. We shall study the zero chemical potential Blume-Capel model for the following choice of the parameters: the magnetic field h and the torus Λ are such that 0 < h < 1, 2/h is not integer, and |Λ| ≥ 49/h 4 finite, where, for any positive real a, we let a be the largest integer smaller than or equal to a.
The time evolution of the model is defined by the Metropolis Markov chain σ t with t = 0, 1, . . . the discrete time variable, see Section 2.2, with Hamiltonian H and connectivity matrix q(σ, η) := 0 if σ, η differ at more than one site 1/(2|Λ|) otherwise .
Given V ⊂ Λ and σ ∈ X , we let σ V be the restriction of σ to V , namely, We let u ∈ X to be the configuration such that u(i) = +1 for all i ∈ Λ. Other two very relevant configurations are d and 0, that is the configuration in which all the spin are minus one and the one in which all the spins are zero. Note that in these configurations the exchange part of the energy is minimal, although the magnetic part is not.
We now define the critical length of the model as  We denote by P c the set of configurations in which all the spins are minus excepted those, which are zeros, in a rectangle of sides long c and c − 1 and in a site adjacent to one of the longest sides of the rectangle (see Fig. 4.2). We denote by Q c the set of configurations in which all the spins are zeros excepted those, which are pluses, in a rectangle of sides long c and c − 1 and in a site adjacent to one of the longest sides of the rectangle (see the caption of Fig. 4.2). We have: We then set A simple direct computation shows that for h small one has Γ c ∼ 4/h. In order to give a result in the spirit of Theorem 2.9 in the case of the Blume-Capel model, we first have to prove preliminary Lemmas ensuring that the Conditions assumed in the general discussion in Section 2.5 are satisfied in the Blume-Capel case. Lemma 4.14 With the parameters chosen as below (4.1), we have that there exists κ > 0 and β 0 > 0 such that for any β > β 0 The Lemmas 4.13 and 4.14 will be proven in Section 4.2 below. The proof of the Lemma 4.15 will be given for completeness in Section 4.2; but we stress that a completely analogous result has been already proven in [ Theorem 4.17 With the parameters chosen as below (4.1), we have that The proof of the theorems is achieved by applying the general results discussed in Section 2.6 and the model dependent lemmas given above. Indeed, Theorem 4.16 follows by Theorem 2.8 and Lemmas 4.13-4.15, whereas Theorem 4.17 follows by Theorem 2.9, and Lemmas 4.13-4.15.

Some more notation
In this section we collect some definitions that will be used in the proof of the Lemmas 4.13-4.15. We let Λ s (σ) := {x ∈ Λ : σ(x) = s} (4.7) for any σ ∈ X with s ∈ {−1, 0, +1}. Recall L denotes the side length of the squared lattice Λ. Let x = (x 1 , x 2 ) ∈ Λ; for 1 , 2 positive integers we let R x 1 , 2 be the collection of the sites (x 1 + n 1 ) mod L, (x 2 + n 2 ) mod L for n i = 0, . . . , x i + i − 1 where i = 1, 2. Roughly speaking, R x 1 , 2 is the rectangle on the torus of side lengths 1 and 2 drawn starting from x and moving in the positive direction along the two coordinate axes. For a positive integer we let Q x := R x , . We denote with R 1 , 2 the set of the configurations σ ∈ X which are rectangular droplet of zeroes with side lengths 1 and 2 in a sea of minus, with 1 , 2 integers such that 2 ≤ 1 , 2 ≤ L − 1. More precisely, σ ∈ R 1 , 2 if and only if there exists x ∈ Λ such that either Λ 0 (σ) = R x 1 , 2 and the spins in the complementary set Λ \ Λ 0 (σ) are negative. Moreover, we denote with R x 1 , 2 ⊂ R 1 , 2 the rectangular droplet of zeroes with side lengths 1 and 2 in a see of minuses and with the lower-left corner in x.
Given a rectangular droplet in R x 1 , 2 , we let N, E, S, and W represent respectively the north, east, south, and west side of the rectangular droplet. For D ∈ {N, E, S, W }, we denote by C x 1 , 2 (n; D) the configuration obtained by adding a zero protuberance of length n to the D-side of the rectangular droplet (see Fig. 4

.3). Note that n is a not negative integer bounded by
Moreover, we denote with C x 1 , 2 (n) the configuration obtained by adding a zero protuberance of length n to any of the four side of the rectangular droplet. (i.e., C x 1 , 2 (n) = D C x 1 , 2 (n; D)). We also let C 1 , 2 (n) = x∈Λ C x 1 , 2 (n). Given a configuration in C x 1 , 2 (n; D) its rectangular envelope is the configuration obtained by flipping to zero the minuses on the side occupied by the protuberance. Then, we have that the rectangular envelope of a configuration in C x 1 , 2 (n; D) belongs to either R 1 +1, 2 or Given a configuration in σ 0 ∈ C 1 , 2 (n; D) with n ≥ 1 a standard growing path is a path (σ 0 , σ 1 , . . . , σ k ) such that σ i+1 is obtained by enlarging by one zero spin the protuberance in σ i and σ k is the rectangular envelope of σ 0 . Note that k = 1 − n and k = 2 − n if D ∈ {E, W } or D ∈ {N, S}, respectively.
Given a configuration in σ 0 ∈ C 1 , 2 (n; D) with n ≥ 1 a standard shrinking path is a path (σ 0 , σ 1 , . . . , σ k ) such that σ i+1 is obtained by flipping to minus one of the zero spins of the protuberance having at most two neighboring minuses and σ k is the configuration obtained by flipping to minus all the spin in the protuberance of σ 0 .
In case of a stripe winding around the torus, i.e., 1 ∨ 2 = L, we use the same notation adopted for the rectangular droplets: R 1 , 2 is the set of the σ ∈ X which are either horizontal stripes of zeroes in a see of minus if 1 = L or vertical stripes if 2 = L. Moreover, we denote with C x 1 , 2 (n) the configuration obtained by adding a zero protuberance of length n to any of the two sides of the stripe with length smaller than L.
For any s ∈ {−1, 0, +1} and x ∈ Λ, we define the spin-flip operator S x s : X → X by letting S x s σ be the configuration such that Recall the definition of path in the configuration space given just above (2.7). A path ω = (ω 1 , . . . , ω m ) ∈ X m is downhill if and only if H(ω i ) ≥ H(ω i+1 ) for any i = 1, . . . , m − 1. We say that a configuration σ ∈ X is a local minimum of the Hamiltonian if and only if H(η) ≥ H(σ) for any η ∈ X such that (σ, η) ∈ Q. For our purposes it is useful to introduce the notion of strict downhill path by saying that a path ω = (ω 1 , . . . , ω m ) ∈ X m is strict downhill if and only if H(ω i ) > H(ω i+1 ) for any i = 1, . . . , m − 1 and σ m is a local minimum of the Hamiltonian.   The proof of Lemma 4.14 needs the discussion of some preliminary results aimed to describe the paths followed by the system when it performs the transition from d to 0. The first step is that of computing energy differences between configurations differing for a single spin. Such a difference will depend only on the configuration in a cross-shaped neighborhood centered at the site with differing spins. Thus, for any x ∈ Λ, we denote with V (x) the neighborhood of x defined as V (x) = {y ∈ Λ : d(x, y) ≤ 1}, where d(·, ·) is the Euclidean distance on the torus. Given a configuration σ, the effect on the Hamiltonian of a change of the spin at site x will depend only on the configuration σ V (x) obtained by restricting σ to V (x) (see, the definition of restriction given above (4.2)). All the possible cases are summarized in the Table 4.2, where the configurations A 1 , A 2 , A 3 , B 1 , . . . , O 3 are listed and the corresponding difference of energies are reported.
We state and prove now the following Lemma regarding enlarging a protuberance of a rectangular droplet following the energy drift.

Lemma 4.18
Given σ ∈ C 1 , 2 (1), any strict downhill path started at σ is either a standard shrinking or growing path.
In the first case ω 2 ∈ R 1 , 2 . Since (ω 2 ) V (y) ∈ {A 1 , B 1 , D 2 , G 2 , I 2 } for any y ∈ Λ, by Table 4.2 it follows that ω 2 is a local minimum of the Hamiltonian. Hence, in this first case, the standard shrinking sequence is found.
In the second case, i.e., when the protuberance is enlarged, for any y ∈ Λ, (ω 2 ) V (y) ∈ {A 1 , B 1 , D 1 , D 2 , G 2 , I 2 }. Notice that there are not anymore neighbors of type B 2 , because the protuberance, now, is at least wide two sites. Hence, if we use the same argument used for ω 1 , we deduce that the only way of lowering the energy is by enlarging the protuberance along the side. We can repeat the same argument until all the line is filled. In this way the standard growing sequence is found.
In case of stripes, i.e., 1 ∨ 2 = L, the proof is almost identical, with only minor adjustments: for instance in the set (4.9) is not present anymore the neighborhood D 2 .
Recall the definition of external boundary given in Section 2.3 and note that in the Blume-Capel case the external boundary of a subset of the configuration space is made of all those configurations not belonging to such a set and such that by changing the value of one single spin the configuration that is obtained belongs to the set. We call a nonempty set C ⊂ S a cycle if it is either a singleton or a connected set such that For any rectangular droplet or stripe ζ ∈ R 1 , 2 we define the cycle made of all the configurations that can be reached starting from ζ via a path whose energy stays below H(ζ)+2−h. The following lemma gives a precise characterization of the minima of the energy of the external boundary of the cycle A ζ for any ζ ∈ R 1 , 2 with 1 , 2 ≥ c , see (4.2).
Proof. Item i). Let us start considering the case 1 ∨ 2 < L, i.e., a proper rectangular droplet. Let ζ ∈ R 1 , 2 . As we have noted in the proof of Lemma 4.18 we have that ζ V (i) ∈ {A 1 , B 1 , D 2 , G 2 , I 2 }, for any i ∈ Λ. This implies that (see Table 4.2) the rectangular droplet ζ is a local minimum of the Hamiltonian. Moreover, by using the results in the table one has that H(S i s ζ) − H(ζ) ≥ 2 − h for any i ∈ Λ excepted for the case of the corner erosion, namely, for i equal to one of the four sites such that ζ V (i) = D 2 and s = −1. In these cases H i (S i s ζ) − H i (ζ) = h. Hence, all the flips but the corner erosion yield a configuration outside A ζ . In particular, we remark that the equality 2−h is attained by adding a protuberance to the rectangular configuration, i.e., S i 0 ζ ∈ C 1 , 2 (1). This implies that C 1 , 2 (1) ⊆ argmin η∈∂A ζ H(η). Let us give the following definitions. Given a configuration η, we define the 0-rectangular envelope the configuration R(η) such that Λ 0 (R(η)) is the smallest rectangle containing Λ 0 (η), where Λ 0 (·) is defined in (4.7). Furthermore, we will call corner-erosion the spin flip from 0 to −1 in the site i with neighborhood V (i) of type D 2 . Moreover, we define the set of configurations: in words, Ξ 1 is the set of all the configurations obtained from ζ by one step of corner-erosion. We note that for any configuration σ ∈ Ξ 1 , i ∈ Λ we have that σ V (i) ∈ {A 1 , B 1 , D 1 , D 2 , G 2 , I 2 }. By inspection of Table 4 and S i −1 σ ∈ A ζ , that gives a second corner-erosion. This suggests the following definition, for any k ≥ 2: In words Ξ k is the set of all the configurations obtained by ζ with k corner-erosions. Notice Again, by inspection of Table 4.2, it follows that, for i ∈ Λ such that γ V (i) ∈ {A 1 , B 1 , D 1 , G 2 , I 2 }: , while for s = −1 we have: In words, from γ ∈ Ξ k any spin-flip that is not corner-erosion increses the energy by more than 2 − h. Using similar arguments as above, for any γ ∈ Ξ c−2 and for any i ∈ Λ we have that γ V (i) ∈ {A 1 , B 1 , D 1 , D 2 , G 2 , I 2 }. If γ V (i) ∈ {A 1 , B 1 , D 1 , G 2 , I 2 }, we get: . In words, from γ ∈ Ξ c−2 any spin-flip, including the corner-erosion, increases the energy by more than 2 − h. This conclude the proof of item i), since we proved that C 1 , 2 (1) ⊆ argmin η∈∂A ζ H(η) and for any other path the configuration reached when exiting A ζ does not belong to argmin η∈∂A ζ H(η).
Item ii). Starting from ζ by c − 2 corner-erosions, it is not possible to change the rectangular envelope, since 1 and 2 are not smaller than c . Furthermore, for any γ ∈ Ξ k , cns-pta2016.tex -14 settembre 2016 25 0:45 with k ≤ c − 2, every sites belonging to the rectangular envelope Λ 0 (R(γ)) is such that there are at least other two neighboring sites with spin zero. By the proof of item i), we have that starting from ζ, all the possible spin flipped configurations S j s ζ for j ∈ Λ and s ∈ {−1, 0, +1}, belongs either to Ξ 1 or to A c ζ . Furthermore, for any η ∈ Ξ 1 , all the possible spin flipped configurations S i s η for i ∈ Λ and s ∈ {−1, 0, +1}, belongs either to Ξ 2 ∪ {ζ} or to A c ζ . For any η ∈ Ξ k , with k ≤ c − 3, all the possible spin flipped configurations S i s η, belongs either to Ξ k+1 ∪ Ξ k+2 or to A c ζ . Finally, for any η ∈ Ξ c−2 , all the possible spin flipped configurations S j s η, belongs either to Ξ c−3 or to A c ζ . Putting together the previous statements, we can analyze all the paths joining ζ to η with η ∈ A ζ and we get that ω k ∈ c−2 k=1 Ξ k ∪ ζ, with k ≤ c − 2. Therefore in any line and row of Λ 0 (η) there are at lest two sites with spin 0. Thus, we have that In case of stripes, i.e., 1 ∨ 2 = L, the proof is simpler. In fact it is not possible anymore to erode a corner: we have that ζ V (i) ∈ {A 1 , B 1 , G 2 , I 2 }, for any i ∈ Λ. This implies that (see Table 4.
Hence, all the flips yield a configuration outside A ζ . Proof. The Lemma can be proved by following the same strategy used in the proof of Lemma 4.19.
We finally come to the proof of Lemma 4.14. Our strategy is similar to the one used in [Lemma 3.13, [23]]. We let T 0 (d) be the collection of the following trivial and non-trivial pairwise disjoint cycles (see Fig. 4.4): 2. all the configurations in P c , 3. the non-trivial cycles A ζ for ζ ∈ R 1 , 2 such that min{ 1 , 2 } ≥ c and max{ 1 , 2 } ≤ L − 2; 4. any configuration obtained by adding a 0 protuberance of length to one side of ζ ∈ R 1 , 2 with min{ 1 , 2 } ≥ c and max{ 1 , 2 } ≤ L − 3, and such that 1 − ≥ c (respectively 2 − ≥ c ) if the protuberance has been added to the side with length 1 (respectively 2 ); 5. any configuration obtained by flipping to zero the spins associated with any nearest neighbor connected 3 subset of the set of sites adjacent to the shortest sides of any rectangular droplet ζ ∈ R 1 , 2 such that c ≤ min{ 1 , 2 } ≤ L − 3 and max{ 1 , 2 } = L − 2; 6. any configuration obtained by flipping to zero the spins associated with any nearest neighbor connected subset of the set of sites adjacent to either the horizontal or vertical sides of any rectangular droplet ζ ∈ R 1 , 2 such that 1 = 2 = L − 2; 7. any configuration obtained by adding an protuberance to one of the two sides of the stripe ζ ∈ R 1 , 2 such that 1 = L and c ≤ 2 ≤ L − 3; 8. any configuration different from 0 obtained by flipping to zero the spins associated with any nearest neighbor connected subset of the set of sites associated with minus one spis of any rectangular droplet ζ ∈ R 1 , 2 such that 1 = L and 2 = L − 2; 9. all the configurations in P c , namely those configuration obtained by adding a 0 protuberance of length one the side long c − 1 of any rectangle ζ ∈ R c−1, c ; 3 A subset A of the torus Λ is said to be nearest neighbor connected if and only if for any x, y ∈ A there exists a sequence of pairwise nearest neighbor sites of A such that the first is x and the last is y. 10. the non-trivial cycles A ζ for ζ ∈ R c−1, c+1 .
The T 0 (d) satisfies the following properties: 0 / ∈ T 0 (d), u / ∈ ∂T 0 (d) ∪ T 0 (d), there exists a cycle C ∈ T 0 (d) such that 0 ∈ B(C), and for any cycle C ∈ T 0 (d) where it is useful to emphasize that T 0 (d) is a collection of pairwise disjoint cycles. The first two properties are immediate by definition. We comment briefly on the last two. At step 6 in the definition of T 0 (d) we added in particular the configurations in which all the spins on the lattice are equal to zero but one which is equal to minus one. It is immediate to realize that the principal boundary of such a trivial cycle is the set {0}. Finally, to prove equation (4.13) one has to examine all the cycles included in T 0 (d) and prove that their principal boundaries are subset of the right hand side of equation (4.13). With the same labelling used in the costruction of the set T 0 (d) we have: 1. by the methods of proof of [14,Lemma 4.12] we have that B(D 0 (d)) = P c ∪P c . 2. for any ζ ∈ P c the set B(ζ) is made of the configurations obtained by enlarging by one site the protuberance; those configurations belong to the not trivial cycles added at step 3. 3. from item i) in Lemma 4.19 it follows that the principal boundary of the cycle A ζ is made by the configurations obtained by adding a protuberance to one of the four sides of the rectangle ζ; these configurations are added at steps 4 and 5; 4. consider the configuration ζ and suppose the protuberance is on the side of length 1 (the argument is analogous in the other case), then B(ζ) is made of the configurations obtained by enlarging by one site the protuberance; those configurations are either trivial cycles added to T 0 (d) at step 4 or elements of the not trivial cycles added at step 3. 5. consider the configuration ζ and assume that it is not a stripe wind around the torus, then B(ζ) is made of the configurations obtained by flipping to zero any minus spin with precisely two neighboring zeros; those configurations are added to T 0 (d) at step 5. On the other hand, if ζ is a stripe winding around the torus the principal boundary is the set of configurations obtained by adding a zero protuberance; those configurations are added to T 0 (d) at step 7. The discussion of the other cases is similar. We just discuss explicitly the case 10: by Lemma 4.20 we have that the principal boundary of the cycle A ζ is made of the configurations obtained by flipping to minus all the spins but one spin of one of the two shortest sides of the rectangular droplet ζ; those configurations are elements of P c . Proof. We have that where in the first equality we have used the fact that for each cycle C in T 0 (d) the principal boundary B(C) is contained in T 0 (d); the first inequality follows from the strong Markov property and the second inequality is a consequence of [25,Theorem 6.23].
Aknowledgements. ENMC thanks ICMS (TU/e, Eindhoven), Eurandom (TU/e, Eindhoven), the Mathematics Department of Delft University, and the Mathematics Department of Utrecht University for kind hospitality. FRN and CS thank A. Bovier for many stimulating discussions. ENMC and FRN thank E. Scoppola and F. den Hollander for illuminating discussions. The authors thank A. Gaudilliere and M. Slowik for useful discussions and comments and an anonymous referee, whose useful comments considerably improved the quality of the manuscript.

A. General bounds
In this appendix we summarize some general results whose statement and proof can already be found in the literature but, sometimes, in slightly different contexts.
for all β large enough. For any pair u, v ∈ X such that u ∈ K(Y, Z) and v ∈ X \ K(Y, Z), we have that H(u) + ∆(u, v) ≥ Φ(Y, Z). In fact, if by absurdity it were H(u) + ∆(u, v) < Φ(Y, Z), it would be possible to construct a path ω starting at v and ending in Y such that Φ ω < Φ(Y, Z), which is in contradiction with v ∈ X \ K(Y, Z). Hence, by (2.14) and (2.15) Recalling (2.2) and (2.3), we get that there exists C such that for β large enough Finally, the upper bound in (1.1) follows from the fact that H(u) + ∆(u, v) ≥ Φ(Y, Z) for any u ∈ K(Y, Z) and v ∈ X \ K(Y, Z).
As regards the lower bound, it can be obtained by picking any path ω = (ω 0 , ω 1 , . . . , ω n ) that realizes the minimax in Φ(Y, Z) and ignore all the transitions that are not the path and using the same argument as in the proof of [Lemma 3.1.1, [7]]. An alternative proof can be given applying the Berman-Konsowa lemma [Proposition 2.4, [8]] which provides a complementary variational principle, in the sense that any test flow will give a lower bound. Hence, the lower bound can be obtained by picking any path ω = (ω 0 , ω 1 , . . . , ω n ), with ω 0 ∈ Y , ω n ∈ Z, such that it realizes the minimax in Φ(Y, Z) and such that H(ω i ) + ∆(ω i , ω i+1 ) ≤ Φ(Y, Z) for i ∈ {0, . . . , n − 1} (recall (2.7) and (2.10)). If we choose a unitary flow for the edges in the path ω and null otherwise, the induced Markov chains is a deterministic chain along the path, so that the expectation in [Proposition 2.4, [8]] is just the contribution of the deterministic path. Hence, for the chosen flow we have: where in the last inequality we used (2.2) and (2.3).

B. Capacity estimate for Metropolis dynamics
In this section we state and prove a slightly more general theorem than Theorem 6.19 of [21]. As in [21], the theorem holds for the Metropolis dynamics introduced in Section 2.2, but in the more general setting of two metastable configurations. We assume that Condition 2.3 holds so that the energy landscape is such that there exist three states x 2 , x 1 , and x 0 such that X s = {x 0 }, X m = {x 1 , x 2 }, and K(x 2 ) > K(x 1 ). This theorem gives indeed the capacity between a configuration x ∈ {x 2 , x 1 } and a set A = {x 0 , x 1 } \ {x} in terms of the energy and cardinality of the minimal gates. Hence, the theorem considers the two case x = x 2 , A = {x 1 , x 0 } and x = x 1 , A = {x 0 }. For x and A as above we let Q x := {y ∈ X : Φ(y, x) < Φ(x, A)} and Q A := {y ∈ X : Φ(y, A) < Φ(x, A)}. By Definition 2.1 and Condition 2.3 we have that Q A is a cycle and Q A ∩ Q x = ∅.
Before stating the theorem we recall first some notions introduced in [21]. Let z ∈ X and B ⊂ X. We say that S(z, B) ⊂ X is the set of saddles for the pair z and B if and only if K(z) = Φ(z, B) for any z ∈ S(z, B). We say that a set W ⊆ S(z, B) is a gate for the pair z and B if for any path ω ∈ Ω(z, B) such that Φ ω = Φ(z, B) we have that W ∩ ω = ∅. A gate W for the pair z and B is said to be minimal if and only if for any proper subset W of W there exists a path joining z to B with maximal height equal to the communication height between z and B which does not pass through W .
cns-pta2016.tex -14 settembre 2016 Theorem 2.1 In the setup introduced above, consider x ∈ {x 2 , x 1 } and A = {x 0 , x 1 } \ {x}. Assume that the minimal gate W for x and A is unique and that for any y ∈ W and w / ∈ Q x ∪ Q A we have p(y, w) = o(1). Then wherep(z) := w∈Qx p β (z, w) andp(z) := w∈Q A p β (z, w).
Proof. Upper bound: by (2.17), [21,Theorem 5.4] and using the strong Markov property, we can write: In fact we have × P x (τ z < τ A , τ z < τ W\z , τ z < τ x )[1 + o(1)] = µ β (x) z∈W P z (τ A < τ x )P x (τ z < τ A , τ z < τ W\z , τ z < τ x )[1 + o(1)] where in the last step we used the fact that: For the second term of 2.2, we have, for any z ∈ W P z (τ A < τ x ) = y∈Q A p(z, y)P y (τ A < τ x ) + y∈Qx p(z, y)P y (τ A < τ x ) + p(z, z)P z (τ A < τ x ) [ where in the first equality we used the fact that p(z, y) = o(1) for any y ∈ (Q x ∪Q A ∪{z}) c ; in the last step, we have used the trivial relation P y (τ A < τ x ) ≤ 1 for bounding the first term of the sum, while the upper bound for the second term follows from P y (τ A < τ x ) ≤ P y (τ ∂Qx < τ x ) and from the recurrence of non trivial cycle (i.e., by Theorem 6.3 in [25], for any y ∈ Q x we have P y (τ ∂Qx < τ x ) = o(1)). Hence, by (2.3) and (2.2) we have: where in the second step we used reversibility and in the third one: and the properties of the cycles Q x and Q A .
Lower bound. In order to prove the lower bound, we adapt to our setting the arguments of the proof in [9, Lemma 3.2]. We consider a subgraph ∆ in the space of configurations obtained removing all the connections to the configurations in ∂Q x \ W. We denote with P x the probability along the trajectories of this restricted process started at x. Hence, by Rayleigh's shortcut rule (see [9, Lemma 4.2]) we have: If we now exploit the property of the restricted process we can write: where we used (2.17) and the strong Markov property. By reversibility, (2.4) becomes Now, the first factor in (2.5) can be rewritten as P z (τ x < τ W ) ≥ y∈Qxp β (z, y) P y (τ x < τ W ) = We want to prove that inside Q x the equilibrium potentialh x,W (y) is exponentially close to 1. Therefore, for any y ∈ Q x , and for δ = Φ(x, A) − K(y) > 0: 1 −h x,W (y) =h W,x (y) ≤ cap β (y, W) cap β (y, x) ≤ C e −βΦ(y,W) e −βΦ(y,x) = C e −βΦ(x,A) e −β(Φ(x,A)−δ) = C e −βδ where in the first inequality we used Proposition 1.2, and in the second inequality Proposition 1.1. Therefore where in the last step, we used the fact that the one step transition probabilitiesp(z, ·) are equal, by construction, to p(z, ·) of the original chain, for any z ∈ W. For the second factor P z (τ A < τ x ): We want to prove that inside Q A the equilibrium potentialh A,x (y) is exponentially close to 1. Therefore for any y ∈ Q A , by using Proposition 1.2, and Proposition 1. Hence, by (2.5), (2.6), and (2.7), (2.1) follows.