Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of positive external magnetic field. In this energy landscape there are $1$ stable configuration and $q-1$ metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as $\beta\to\infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We also geometrically identify the union of all minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration.


Introduction
Metastability is a phenomenon that is observed when a physical system is close to a first order phase transition.More precisely, this phenomen takes place when the physical system, for some specific values of the parameters, is imprisoned for a long time in a state which is different from the equilibrium state.The former is known as the metastable state, the latter is the stable state.
After a long (random) time, the system may exhibit the so-called metastable behavior and this happens when the system performs a sudden transition from the metastable state to the stable state.On the other hand, when the system lies on the phase coexistence line, it is of interest to understand precisely the transition between two (or more) stable states.This is the so-called tunneling behavior.
The phenomenon of metastability occurs in several physical situations, such as supercooled liquids, supersaturated gases, ferromagnets in the hysteresis loop and wireless networks.For this reason, many models for the metastable behavior have been developed throughout the years.In these models a suitable stochastic dynamics is chosen and typically three main issues are investigated.The first is the study of the first hitting time at which the process starting from a metastable state visits some stable states.The second issue is the study of the so-called set of critical configurations, which are those configurations that the process visits during the transition from the metastable state to some stable states.The third issue is the study of the tube of typical paths, i.e., the set of the typical trajectories followed by the process during the transition from the metastable state to some stable states.On the other hand, when a system exhibits tunneling behavior the same three issues above are investigated for the transition between any two stable states.
In this paper we study the metastable behavior of the q-state Potts model with non-zero external magnetic field on a finite two-dimensional discrete torus Λ.We will refer to Λ as a grid-graph.The q-state Potts model is an extension of the classical Ising model from q = 2 to an arbitrary number q of spins with q > 2. The state space X is given by all possible configurations σ such that at each site i of Λ lies a spin with value σ(i) ∈ {1, . . ., q}.To each configuration σ ∈ X we associate an energy H(σ) that depends on the local ferromagnetic interaction, J = 1, between nearest-neighbor spins, and on the external magnetic field h related only to a specific spin value.Without loss of generality, we choose this spin equal to the spin 1.We study the q-state ferromagnetic Potts model with Hamiltonian H(σ) in the limit of large inverse temperature β → ∞.The stochastic evolution is described by a Glauber-type dynamics, that is a Markov chain on the finite state space X with transition probabilities that allow single spin-flip updates and that is given by the Metropolis algorithm.This dynamics is reversible with respect to the stationary distribution that is the Gibbs measure µ β , see (2.3).
Our analysis focuses on the model to which we will refer as q-Potts model with positive external magnetic field.In this energy landscape there are q − 1 degenerate-metastable states and only one stable state.In the metastable configurations all spins are equal to some m, for m ∈ {2, . . ., q}, while the stable state is the configuration in which all spins are equal to 1.In this case, we focus our attention on the transition from one of the metastable states to the stable configuration.
The goal of this paper is to investigate all the three issues of metastability for the metastable behavior of the q-Potts model with positive external magnetic field.More precisely, we investigate the asymptotic behavior of the transition time, and identify the set of critical configurations and the tube of typical trajectories for the transition from a metastable state to the unique stable state.Furthermore, we also identify the union of all the critical configurations for the transition from a metastable configuration to the other metastable states.
Let us now briefly describe the strategy that we adopt.First we show that the metastable set contains the q − 1 configurations where all spins are equal to some m ∈ {2, . . ., q}.We give asymptotic bounds in probability for the first hitting time from any metastable state to the stable configuration and we identify the order of magnitude of the expected hitting time.Moreover, we characterize the behavior of the mixing time in the low-temperature regime and give a lower and un upper bound of the spectral gap.Finally, we find the set of all minimal gates for the transition from a metastable state to the stable configuration.For any m ∈ {2, . . ., q}, if the starting configuration is the one with all spins equal to m, we prove that the minimal gate contains those configurations in which all spins are m except those, which are 1, in a quasi-square with a unit protuberance on one of the longest sides.Furthermore, we prove that during the metastable transition the process almost surely does not visit any metastable state different from the initial one, and we exploit this result to identify the union of all minimal gates for the transition from a metastable state to the other metastable configurations.Finally, we identify geometrically those configurations that belong to the tube of typical paths for the transition from any metastable state to the stable state.
The Potts model is one of the most studied statistical physics models, as the vast literature on the subject attests, both on the mathematics side and the physics side.The study of the equilibrium properties of the Potts model and their dependence on q, have been investigated on the square lattice Z d in [7,6], on the triangular lattice in [8,39] and on the Bethe lattice in [2,32,36].The mean-field version of the Potts model has been studied in [30,37,38,42,60].Furthermore, the tunneling behaviour for the Potts model with zero external magnetic field has been studied in [52,11,47].In this energy landscape there are q stable states and there is not any relevant metastable state.In [52], the authors derive the asymptotic behavior of the first hitting time for the transition between stable configurations, and give results in probability, in expectation and in distribution.They also characterize the behavior of the mixing time and give a lower and an upper bound for the spectral gap.In [11], the authors study the tunneling from a stable state to the other stable configurations and between two stable states.In both cases, they geometrically identify the union of all minimal gates and the tube of typical trajectories.Finally, in [47], the authors study the model in two and three dimensions.In both cases, they give a description of gateway configurations that is suitable to allow them to prove sharp estimate for the tunneling time by computing the so-called prefactor.These gateway configurations are quite different from the states belonging to the minimal gates identified by [11].The q-Potts model with non-zero external magnetic field has been studied in [12], where the authors study the energy landscape defined by a Hamiltonian function with negative external magnetic field.In this scenario there are a unique metastable configuration and q − 1 stable states, and the authors answer to all the three issues of the metastability introduced above for the transition from the metastable state to the set of the stable states and also to any fixed stable state.Furthermore, they give sharp estimates on the expected transition time by computing the prefactor.
State of art All grouped citations here and henceforth are in chronological order of publication.In this paper we adopt the framework known as pathwise approach, that was initiated in 1984 by Cassandro, Galves, Olivieri, Vares in [20] and it was further developed in [56,57,58] and independtly in [21].The pathwise approach is based on a detailed knowledge of the energy landscape and, thanks to ad hoc large deviations estimates it gives a quantitative answer to the three issues of metastability which we described above.This approach was further developed in [49,24,25,53,40,41] to distinguish the study of the transition time and of the critical configurations from the study of the third issue.This is achieved proving the recurrence property and identifying the communication height between the metastable and the stable state that are the only two model-dependent inputs need for the results concerning the first issue of metastability.In particular, in [49,24,25,53,40,41] this method has been exploited to find answers valid increasing generality in order to reduce as much as possible the number of model dependent inputs necessary to study the metastable and tunneling behaviour, and to consider situations in which the energy landscapes has multiple stable and/or metastable states.For this reason, the pathwise approach has been used to study metastability in statistical mechanics lattice models.The pathwise approach has been also applied in [4,22,29,48,51,54,55,58] with the aim of answering to the three issues for Ising-like models with Glauber dynamics.Moreover, it was also applied in [45,33,44,3,53,61] to study the transition time and the gates for Ising-like and hard-core models with Kawasaki and Glauber dynamics.Furthermore, this method was applied to probabilistic cellular automata (parallel dynamics) in [23,26,27,59,31].
The pathwise approach is not the only method which is applied to study the physical systems that approximate a phenomenon of metastability.For instance, the so-called potentialtheoretical approach exploits a suitable Dirichlet form and spectral properties of the transition matrix to investigate the sharp asymptotic of the hitting time.An interesting aspect of this method is that it allows to estimate the expected value of the transition time including the so-called prefactor, i.e., the coefficient that multiplies the leading-order exponential factor.To find these results, it is necessary to prove the recurrence property, the communication height between the metastable and the stable state and a detailed knowledge of the critical configurations as well of those configurations connected with them by one step of the dynamics, see [17,18,15,28].In particular, the potential theoretical approach was applied to find the prefactor for Ising-like models and the hard-core model in [5,19,28,16,34,46,35] for Glauber and Kawasaki dynamics and in [50,13] for parallel dynamics.Recently, other approaches have been formulated in [9,10,43] and in [14] and they are particularly adapted to estimate the pre-factor when dealing with the tunnelling between two or more stable states.
Outline The outline of the paper is as follows.In Section 2 we define the ferromagnetic q-state Potts model and the Hamiltonian that we associate to each Potts configuration.In Section 3 we give a list of both model-independent and model dependent definitions that occur to state our main results in Section 4. In Section 5 we analyse the energy landscape and give the explicit proofs of the main results stated in Subsections 4.1 and 4.2.Subsection 6.2 is devoted to the study on the transition from a metastable state to the other metastable configurations.In Subsections 6.3 and 6.4 we give the explicit proofs of the main results on the critical configurations and on the tube of typical paths, respectively.

Model description
In the q-state Potts model each spin lies on a vertex of a finite two-dimensional rectangular lattice Λ = (V, E), where V = {0, . . ., K − 1} × {0, . . ., L − 1} is the vertex set and E is the edge set, namely the set of the pairs of vertices whose spins interact with each other.We consider periodic boundary conditions.More precisely, we identify each pair of vertices lying on opposite sides of the rectangular lattice, so that we obtain a two-dimensional torus.Two vertices v, w ∈ V are said to be nearest-neighbors when they share an edge of Λ.We denote by S the set of spin values, i.e., S := {1, . . ., q} and assume q > 2. To each vertex v ∈ V is associated a spin value σ(v) ∈ S, and X := S V denotes the set of spin configurations.We denote by 1, . . ., q ∈ X those configurations in which all the vertices have spin value 1, . . ., q, respectively.To each configuration σ ∈ X we associate the energy H(σ) given by where J is the coupling or interation constant and h is the external magnetic field.The function H : X → R is called Hamiltonian or energy function.The Potts model is said to be ferromagnetic when J > 0, and antiferromagnetic otherwise.In this paper we set J = 1 without loss of generality and, we focus on the ferromagnetic q-state Potts model with non-zero external magnetic field.More precisely, we study the model with positive external magnetic field, i.e., we rewrite (2.1) by considering the magnetic field h pos := h, The Gibbs measure for the q-state Potts model on Λ is a probability distribution on the state space X given by where β > 0 is the inverse temperature and where Z := σ ′ ∈X e −βHpos(σ ′ ) .
The spin system evolves according to a Glauber-type dynamics.This dynamics is described by a single-spin update Markov chain {X β t } t∈N on the state space X with the following transition probabilities: for σ, σ ′ ∈ X , where [n] + := max{0, n} is the positive part of n and for any σ, σ ′ ∈ X .Q is the so-called connectivity matrix and it is symmetric and irreducible, i.e., for all σ, σ ′ ∈ X , there exists a finite sequence of configurations ω 1 , . . ., ω n ∈ X such that Hence, the resulting stochastic dynamics defined by (2.4) is reversible with respect to the Gibbs measure (2.3).The triplet (X , H, Q) is the so-called energy landscape.
The dynamics defined above belongs to the class of Metropolis dynamics.Given a configuration σ in X , at each step 1. a vertex v ∈ V and a spin value s ∈ S are selected independently and uniformly at random; 2. the spin at v is updated to spin s with probability where σ v,s is the configuration obtained from σ by updating the spin in the vertex v to s, i.e., Hence, at each step the update of vertex v depends on the neighboring spins of v and on the energy difference (2.8)

Definitions and notations
In order to state our main results, we need to give some definitions and notations which are used throughout the next sections.

Model-independent definitions and notations
We now give a list of model-independent definitions and notations that will be useful in formulating our main results.
-We define optimal paths those paths that realize the min-max in (3.2) between σ and σ ′ .Formally, we define the set of optimal paths between σ, σ ′ ∈ X as -For any σ ∈ X , let be the set of states with energy strictly smaller than H(σ).We define stability level of σ the energy barrier If -The bottom F (A) of a non-empty set A ⊂ X is the set of the global minima of H in A, i.e., In particular, X s := F (X ) is the set of the stable states.
-For any σ ∈ X and any A ⊂ X , A = ∅, we set -We define the set of metastable states as We denote by Γ m the stability level of a metastable state.
-We define metastable set at level V the set of all the configurations with stability level larger tha V , i.e., -The set of minimal saddles between σ, σ ′ ∈ X is defined as -We say that η ∈ S(σ, σ ′ ) is an essential saddle if there exists ω ∈ Ω opt σ,σ ′ such that either -We say that W(σ, σ ′ ) is a minimal gate for the transition from σ to σ ′ if it is a minimal (by inclusion) subset of S(σ, σ ′ ) that is visited by all optimal paths.More in detail, it is a gate and for any W ′ ⊂ W(σ, σ ′ ) there exists ω ′ ∈ Ω opt σ,σ ′ such that ω ′ ∩ W ′ = ∅.We denote by G = G(σ, σ ′ ) the union of all minimal gates for the transition σ → σ ′ .
-Given a non-empty subset A ⊂ X and a configuration σ ∈ X , we define as the first hitting time of the subset A for the Markov chain {X β t } t∈N starting from σ at time t = 0.
-Given a non-empty subset A ⊆ X , it is said to be connected if for any σ, η ∈ A there exists a path ω : σ → η totally contained in A. Moreover, we define ∂A as the external boundary of A, i.e., the set When C is a singleton, it is said to be a trivial cycle.Let C (X ) be the set of cycles of X .
-The depth of a cycle C is given by If C is a trivial cycle we set Γ(C) = 0.
-Given a non-empty set A ⊂ X , we denote by M(A) the collection of maximal cycles A, i.e., -For any A ⊂ X , we define the maximum depth of A as the maximum depth of a cycle contained in A, i.e.,

Γ(A) := max
In [53,Lemma 3.6] the authors give an alternative characterization of (3.17) as the maximum initial energy barrier that the process started from a configuration η ∈ A possibly has to overcome to exit from A, i.e., -For any σ ∈ X , if A is a non-empty target set, we define the initial cycle for the transition from σ to A as -A β → f (β) is said to be super-exponentially small (SES) if

Model-dependent definitions and notations
In this section we give some further model-dependent notations, which hold for any fixed q-Potts configuration σ ∈ X .
-For any v, w ∈ V , we write w ∼ v if there exists an edge e ∈ E that links the vertices v and w.
-We denote the edge that links the vertices v and w as (v, w) ∈ E. Each v ∈ V is naturally identified by its coordinates (i, j), where i and j denote respectively the number of the row and of the column where v lies.Moreover, the collection of vertices with first coordinate equal to i = 0, . . ., K − 1 is denoted as r i , which is the i-th row of Λ.The collection of those vertices with second coordinate equal to j = 0, . . ., L − 1 is denoted as c j , which is the j-th column of Λ.
-We define the set C s (σ) ⊆ R 2 as the union of unit closed squares centered at the vertices v ∈ V such that σ(v) = s.We define s-clusters the maximal connected components C s 1 , . . ., C s n , n ∈ N, of C s (σ).-For any s ∈ S, we say that a configuration σ ∈ X has an s-rectangle if it has a rectangular cluster in which all the vertices have spin s.
-Let R 1 an r-rectangle and R 2 an s-rectangle.They are said to be interacting if either they intersect (when r = s) or are disjoint but there exists a site v / ∈ R 1 ∪ R 2 such that σ(v) = r, s and v has two nearest-neighbor w, u lying inside R 1 , R 2 respectively.For instance, in Figure 12(c) the gray rectangles are not interacting.Furthermore, we say that R 1 and R 2 are adjacent when they are at lattice distance one from each other, see for instance Figure 12(d).Since in our scenario there are q types of spin, note that here we give a geometric characterization of interacting rectangles.This geometric definition coincides with the usual one, in which flipping the spin on vertex v decreases the energy, only if the two rectangles have the same spin value.
-We set R(C s (σ)) as the smallest rectangle containing C s (σ).
-Let R ℓ1×ℓ2 be a rectangle in R 2 with sides of length ℓ 1 and ℓ 2 .
-Let s ∈ S. If σ has a cluster of spins s which is a rectangle that wraps around Λ, we say that σ has an s-strip.For any r, s ∈ S, we say that an s-strip is adjacent to an r-strip if they are at lattice distance one from each other.For instance, in Figure 12(a)-(b) there are depicted vertical and horizontal adjacent strips, respectively.
-Ra,b (r, s) denotes the set of those configurations in which all the vertices have spins equal to r, except those, which have spins s, in a rectangle a × b, see Figure 1(a).Note that when either a = L or b = K, Ra,b (r, s) contains those configurations which have an r-strip and an s-strip.
-Bl a,b (r, s) denotes the set of those configurations in which all the vertices have spins r, except those, which have spins s, in a rectangle a × b with a bar 1 × l adjacent to one of the sides of length b, with 1 ≤ l ≤ b − 1, see Figure 1(b).
-We define as the critical length.We color white the vertices whose spin is r and we color gray the vertices whose spin is s.
4 Main results on the q-state Potts model with positive external magnetic field This section is devoted to the statement of our main results on the q-state Potts model with positive external magnetic field.Note that we give the proof of the main results by considering the condition L ≥ K ≥ 3ℓ * , where ℓ * is defined in (3.21).It is possible to extend the results to the case K > L by interchanging the role of rows and columns in the proof.
In this scenario related to the Hamiltonian H pos , we add either a subscript or a superscript "pos" to the notation of the model-independent quantities (defined in general in Subsection 3.1) in order to remind the reader that these quantities are computed in the case of positive external magnetic field.For example, we denote the set of the metastable configurations of the Hamiltonian H pos by X m pos := {η ∈ X : is the stability level related to H pos of any configuration ξ ∈ X .In order to give the main results on the q-Potts model with energy function defined in (2.2), we have the following assumption.Assumption 4.1.We assume that the following conditions are verified: (i) the magnetic field h pos := h is such that 0 < h < 1; (ii) 2/h is not integer.

Energy landscape
Using the definition (2.2) and by simple algebraic calculations, in the following proposition we identificate the set of the global minima of H pos .Proposition 4.1 (Identification of the stable configuration).Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.Then, the set of global minima of the Hamiltonian (2.2) is given by X s pos := {1}.In the next theorem we define the configurations that belong to X m pos and give an estimate of the stability level Γ m pos .We refer to Figure 2 for a pictorial representation of the 4-state Potts model related to the Hamiltonian H pos .Theorem 4.1 (Identification of the metastable states).Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.Then, and, for any m ∈ X m pos , Proof.The theorem follows by [24,Theorem 2.4] since the first assumption follows by Propositions 5.1 and 5.2 and the second assumption is satisfied thanks to Proposition 4.2.Using (4.2), in Subsection 5.3 we prove the following corollary.In the following proposition, that we prove in Subsection 5.2, we investigate on the stability level of any configuration η ∈ X \{2, . . ., q}.Proposition 4.2 (Estimate on the stability level).If the external magnetic field is positive, then for any η ∈ X \{1, . . ., q} and m ∈ {2, . . ., q}, V pos η ≤ 2 < Γ pos (m, X s pos ).Exploiting the estimate of the stability level given in Proposition 4.2, we prove the following result on a recurrence property to set of the metastable and stable configurations, i.e., {1, . . ., q}.Theorem 4.2 (Recurrence property).Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.For any σ ∈ X and for any ǫ > 0, there exists k > 0 such that for β sufficiently large

Asymptotic behavior of the first hitting time to the stable state and mixing time
We are interested in studying the first hitting time X s pos = {1} of the stable configuration starting from any m ∈ X m pos , i.e., τ m X s pos .In the following theorem we give asymptotic bounds in probability and identify the order of magnitude of the expected value of τ m X s pos .Moreover, we identify the exponent at which the mixing time of the Markov chain {X β t } t∈N asymptotically grows as β and give an upper and a lower bound for the spectral gap, see see (3.12) and (3.13).In the literature there exist some model-independent results on the asymptotic rescaled distribution of the first hitting time from some η ∈ X to a certain target G ⊂ X , see for instance [49,Theorem 4.15], [40,Theorem 2.3], [53,Theorem 3.19].Unfortunately none of these results are suitable for our scenario when η ∈ X m pos and G = X s pos .This fact follows by the presence of multiple degenerate metastable states that implies the presence of other deep wells in X different from the initial cycle C m X s pos (Γ m pos ).Hence, we consider a different target and we investigate the asymptotic rescaled distribution of the first hitting time from a metastable state to the subset G ⊂ X setting We defer to Subsection 5.3 for the proof of the following theorem.
Theorem 4.4.Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.Let m ∈ X m pos and let G as defined in (4.5).Then, Note that by definition (4.5), by Proposition 4.1 and by Theorem 4.1, we have F (G) = X s pos and that the maximal stability level is V (G) = Γ m pos .

Minimal gates for the metastable transition
A further goal is to identify the union of all minimal gates for the transition from any metastable state to the unique stable state X s pos = {1}.In order to do this, for any m ∈ S\{1}, let us define We refer to Figure 14(b)-(c) for an example of configurations belonging respectively to W ′ pos (m, X s pos ) and to W pos (m, X s pos ).These sets are investigated in Subsection 6.1.In particular, in Proposition 6.1 we show that W pos (m, X s pos ) is a gate for the transition from any m ∈ X m pos to X s pos .Furthermore, in Subsection 6.3 we prove the following result.
Theorem 4.5 (Minimal gates for the transition from m ∈ X m pos to X s pos ).Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.Then, W pos (m, X s pos ) is a minimal gate for the transition from any metastable state m ∈ X m pos to X s pos = {1}.Moreover, Figure 3: Viewpoint from above of the energy landscape depicted in Figure 2.For every m ∈ X m pos , the cycle whose bottom is the stable state 1 is deeper than the initial cycles C m X s pos (Γ m pos ).These last cycles are depicted with circles whose diameter is smaller than the one related to the stable state 1.
We remark that in [12, Theorems 4.5 and 4.6] the authors identify the union of all minimal gates for the metastable transitions for the q-Potts model with negative external magnetic fields.These minimal gates have the same geometric definition of those of our scenario, the main difference is that in the negative case there are q − 1 possible "colors" for the vertices inside the quasi-square with a unit protuberance.
In the next corollary we show that the process typically intersects W pos (m, X s pos ) during the transition m ∈ X m pos → X s pos .Corollary 4.2.Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.Then, for any m ∈ X m pos Proof.The corollary follows from Proposition 6.1 and from [49,Theorem 5.4].

Minimal gates for the transition from a metastable state to the other metastable configurations
In Subsection 6.2 we study the transition from a metastable state to the set of the other metastable states.We prove that during the transition from any m ∈ X m pos to X s pos almost surely the process does not intersect X m pos \{m}, and we exploit this result to identificate the union of all minimal gates for this type of transition.
We defer the proof of the above theorem in Subsection 6.3.Note that in the negative scenario [12] the theorem corresponding to Theorem 4.6 is not present since there is only one metastable state.

Further model-independent and model-dependent definitions
In addition to the list of Subection 3.1, in order state the main result concerning the tube of the typical trajectories we give some further definitions that are taken from [53], [25] and [58].
-A cycle-path (C 1 , . . ., C m ) is said to be downhill (strictly downhill ) if the cycles C 1 , . . ., C m are pairwise connected with decreasing height, i.e., when -For any C ∈ C (X ), we define as -The relevant cycle where δ is the minimum energy gap between any optimal and any non-optimal path from σ to A.
-We denote the set of cycle-paths that lead from σ to A and consist of maximal cycles in X \A as -Given a non-empty set A ⊂ X and σ ∈ X , we constructively define a mapping G : Ω σ,A → P σ,A in the following way.
We note that ω is a finite sequence and ω n ∈ A, so there exists an index n(ω) ∈ N such that ω m n(ω) = ω n ∈ A and there the procedure stops.
Let J C,A be the collection of all cycle-paths (C 1 , . . ., C m ) that are vtj-connected to A and such that C 1 = C.
-Given a non-empty set A and σ ∈ X , we define ω ∈ Ω σ,A as a typical path from σ to A if its corresponding cycle-path G(ω) is vtj-connected to A and we denote by Ω vtj σ,A the collection of all typical paths from σ to A, i.e., -We define the tube of typical paths T A (σ) from σ to A as the subset of states η ∈ X that can be reached from σ by means of a typical path which does not enter A before visiting η, i.e., Moreover, we define T A (σ) as the set of all maximal cycles that belong to at least one vtj-connected path from Note that and that the boundary of T A (σ) consists of states either in A or in the non-principal part of the boundary of some C ∈ T A (σ): For the sake of semplicity, we will also refer to T A (σ) as tube of typical paths from σ to A.
Furthermore, in addition to the list given in Subsection 3.2, we give some further modeldependent definitions.
-For any m, s ∈ S, m = s, we define -For any m ∈ S\{1}, we define

Main results on the tube of typical trajectories
In this subsection we give our main result concerning the tube of the typical trajectories for the transition m → X s pos for any fixed m ∈ X m pos .The tube of typical paths for this transition turns out to be As illustrated in the next result, which we prove in Section 6.4, T X s pos (m) includes those configurations with a positive probability of being visited by the Markov chain {X t } β t∈N started in m before hitting X s pos in the limit β → ∞.Note that the relation between T X s pos (m) and T X s pos (m) is given by (4.19).Theorem 4.7.If the external magnetic field is positive, then for any m ∈ X m pos the tube of typical trajectories for the transition m → X s pos is (4.23).Furthermore, 5 Energy landscape analysis and asymptotic behavior In this section we analyse the enrgy landscape of the q-state Potts model with positive external magnetic field.First we recall some useful definitions and lemmas from [52].

Known local geometric properties
In the following list we introduce the notions of disagreeing edges, bridges and crosses of a Potts configuration on a grid-graph Λ.
-For any i = 0, . . ., K − 1, let be the total number of disagreeing edges that σ has on row r i .Furthermore, for any j = 0, . . ., L − 1 let be the total number of disagreeing edges that σ has on column c j .
-We define d h (σ) as the total number of disagreeing horizontal edges and d v (σ) as the total number of disagreeing vertical edges, i.e., and (5.4) Since we may partition the edge set E in the two subsets of horizontal edges E h and of vertical edges E v , such that E h ∩ E v = ∅, the total number of disagreeing edges is given by (5.5) -We say that σ has a horizontal bridge on row r if σ(v) = σ(w), for all v, w ∈ r.
-We say that σ has a vertical bridge on column c if σ(v) = σ(w), for all v, w ∈ c.
-We say that σ ∈ X has a cross if it has at least one vertical and one horizontal bridge.
For sake of semplicity, if σ has a bridge of spins s ∈ S, then we say that σ has an s-bridge.
Similarly, if σ has a cross of spins s, we say that σ has an s-cross.-For any s ∈ S, the total number of s-bridges of the configuration σ is denoted by B s (σ).
Note that if a configuration σ ∈ X has an s-cross, then B s (σ) is at least 2 since the presence of an s-cross implies the presence of two s-bridges, i.e., of a horizontal s-bridge and of a vertical s-bridge.
We conclude this section by recalling the following three useful lemmas from [52].These results give us some geometric properties for the q-state Potts model on a grid-graph and they are verified regardless of the definition of the external magnetic field.
Then for every s ∈ S we have that (iii) if σ has no horizontal bridge on row r, then d r (σ) ≥ 2; (iv) if σ has no vertical bridge on column c, then d c (σ) ≥ 2.

Metastable states and stability level of the metastable configurations
By Proposition 4.1, H pos has only one global minimum, X s pos = {1}.Furthermore, the configurations 2, . . ., q are such that H pos (2) = • • • = H pos (q).In this subsection, our aim is to prove that the metastable set X m pos is the union of these configurations.We are going to prove this claim by steps.We begin by obtaining an upper bound for the stability level of the states 2, . . ., q.Given m ∈ {2, . . ., q}, let us compute the following energy gap between any σ ∈ X and m, where in the last equality we used (5.5). (5.8) Proof.The proof proceeds analogously to the proof of [12,Lemma 5.4] by replacing 1 with m, s with 1 and ω with ω.See Appendix A.1.2 for the explicit proof.
In the next lemma we show that for any m ∈ X m pos , B2 ℓ * −1,ℓ * (m, 1) is connected to the stable set X s pos by a path that does not overcome the energy value 4ℓ * − h(ℓ * (ℓ * − 1) + 1) + H pos (m).
Lemma 5.5.Let m ∈ X m pos .If the external magnetic field is positive, then for any σ ∈ B2 ℓ * −1,ℓ * (m, 1) there exists a path γ : σ → m such that the maximum energy along γ is bounded as (5.9) Proof.The proof proceeds analogously to the proof of [12,Lemma 5.5] by replacing 1 with m, s with 1 and ω with ω.See Appendix A.1.3 for the explicit proof.
We are now able to prove the following propositions, in which we give an upper bound and a lower bound for Γ pos (m, X s pos ) := Φ pos (m, X s pos ) − H pos (m), for any m ∈ X m pos .
Proposition 5.1 (Upper bound for the communication height).If the external magnetic field is positive, then for every m ∈ X m pos , (5.10) Proof.The upper bound (5.10) follows by the proof of Lemma 5.5, where we proved that Proof.The proof proceeds analogously to the proof of [12,Lemma 5.6] by replacing 1 with m, s with 1 and ω with ω.See Appendix A.1.5 for the explicit proof.
Let σ ∈ X and let v ∈ V .We define the tile centered in v, denoted by v-tile, as the set of five sites consisting of v and its four nearest neighbors.See for instance Figure 5.A v-tile is said to be stable for σ if by flipping the spin on vertex v from σ(v) to any s ∈ S the energy difference H pos (σ v,s ) − H pos (σ) is greater than or equal to zero.In Lemma 5.7 we define the set of all possible stable tiles induced by the Hamiltonian (2.2) and next we exploit this result to prove Proposition 5.3.For any σ ∈ X , v ∈ V and s ∈ S, we define n s (v) as the number of nearest neighbors to v with spin s in σ, i.e., (5.12) Lemma 5.7 (Characterization of stable v-tiles in a configuration σ).Let σ ∈ X and let v ∈ V .
If the external magnetic field is positive, then the tile centered in v is stable for σ if and only if it satisfies one of the following conditions.
(1) For any m ∈ S, m = 1, if σ(v) = m, v has either at least three nearest neighbors with spin m or two nearest neighbors with spin m and two nearest neighbors with spin r, t ∈ S\{m} such that they may be not both equal to 1, see Figure 5 (2) If σ(v) = 1, v has either at least two nearest neighbors with spin 1, see Figure 5(b),(e),(n)-(q) or it has one nearest neighbor 1 and three nearest neighbors with spin r, s, t ∈ S\{1} different from each other, see Figure 5(s).In particular, if σ(v) = m, then (5.13) Proof.Let σ ∈ X and let v ∈ V .To find if a v-tile is stable for σ we reduce ourselves to flip the spin on vertex v from σ(v) = m to a spin r such that n r (v) > 1.Indeed, otherwise the energy difference (2.8) is for sure strictly positive.Let us divide the proof in several cases.Case 1. Assume that n m (v) = 0 in σ.Then the corresponding v-tile is not stable for σ.Indeed, for any m ∈ S and r / ∈ {1, m}, by flipping the spin on vertex v from m to r we get Moreover, by flipping the spin on vertex v from m = 1 to 1 we have (5.16) Furthermore, for any m = 1, if r = 1 by flipping the spin on vertex v from m to 1 we have Then, in view of the energy difference (2.8), for any m ∈ S and r / ∈ {1, m}, by flipping the spin on vertex v from m to r we have ( Furthermore, by flipping the spin on vertex v from m = 1 to 1 we get Hence, for any m ∈ S, if v has two nearest neighbors with spin m and two nearest neighbors with spin 1, then the corresponding v-tile is not stable.In all the other cases, for any m ∈ S, if v has two nearest neighbors with spin m, the corresponding v-tile is stable for σ, see Figure 5(f)-(q).Case 4. Assume that v ∈ V has three nearest neighbors with spin m and one nearest neighbor r in σ, i.e., n m (v) = 3 and n r (v) = 1.Then, for any m ∈ S and r / ∈ {1, m}, by flipping the spin on vertex v from m to r we have (5.20) Moreover, by flipping the spin on vertex v from m = 1 to 1 we get We are now able to define the set of the local minima M pos and the set of the stable plateaux Mpos of the energy function (2.2).More precisely, the set of local minima M pos is the set of stable points, i.e., M pos := {σ ∈ X : H pos (F (∂{σ})) > H pos (σ)}.While, a plateau D ⊂ X , namely a maximal connected set of equal energy states, is said to be stable if H pos (F (∂D)) > H pos (D).These will be given by the following sets: Proof.A configuration σ ∈ X is said to be a local minimum, respectively a stable plateau, when for any v ∈ V and s ∈ S the energy difference (2.8) is either strictly positive, respectively null.When σ has at least one unstable v-tile, for some v ∈ V , by flipping the spin on vertex v it is possible to define a configuration with energy value strictly lower than H pos (σ).Thus, in this case σ does not belong to M pos ∪ Mpos .Hence, below we give a geometric characterization of any σ ∈ M pos ∪ Mpos under the constraint that for any v ∈ V the corresponding v-tile is stable for σ.In order to do this, we exploit Lemma 5.7 and during the proof we often refer to Figure 5.In fact, a local minimum and a stable plateau are necessarily the union of one or more classes of stable tiles in Figure 5. Hence, we obtain all the local minima by enumerating all the possible ways in which the tiles in Figure 5 may be combined.
Step 1.If σ has only stable tiles as in Figure 5(a)-(b), then there are no interfaces, i.e., no disagreeing edges, and σ ∈ M 1 pos .
Step 2. If σ has only stable tiles as in Figure 5(a)-(e), then either there are not interfaces or there are only either horizontal or vertical interfaces of length L and K, respectively.Thus, σ ∈ M 1 pos ∪ M 2 pos .
Step 3. Assume that σ has only stable tiles as in Figure 5(a)-(g), (n) and (o).For sake of semplicity we proceed by several steps.
Step 3.1.The v-tiles as in Figure 5(n) and (o) are such that any spin-update on vertex v strictly increases the energy of at least h.Henceforth, for the case Figure 5(n) and for any m ∈ S, we get that σ may have some 1-rectangles both in a sea of spins m and inside a strip of spins m.Note that the minimum side of these rectangles must be larger than or equal to two.Indeed, from Lemma 5.7 we know that a v-tile where v has spin 1 and it has one nearest neighbor 1 and three nearest neighbors with spin m is not stable for σ.On the other hand, at this step a stable tile as in Figure 5(o) belongs to the configuration σ ∈ M pos ∪ Mpos if and only if it belongs to a 1-strip between two r-strips.
Step 3.2.Now we focus on the stable tile for σ as in Figure 5(f) and (g).If r = 1, the stable tile as in Figure 5(f) and (g) is not stable for σ by item (1) of Lemma 5.7.It follows that σ has not m-rectangles neither inside a sea of spins 1 nor inside a strip of spins 1.Hence, assume that m, r = 1, r = m.In this case, we have to study the clusters as in Figure 6.
Figure 6: Possible clusters of spins different from 1 by considering the stable tile depicted in Figure 5(f) and (g) for σ centered in the vertex v.We color black the m-rectangle and gray the r-rectangles.
We claim that these types of clusters do not belong to σ ∈ M pos ∪ Mpos regardless of what is around them.Let us begin by considering the cluster depicted in Figure 6(a).We construct a downhill path in which, starting from the vertex v, all the spins m on those ℓ vertices next to a gray rectangle are flipped from m to r, see Figure 6(a).In particular, during the first ℓ − 1 steps of this path, the spin which is flipping from m to r has two nearest neighbors with spin m and two nearest neighbors with spin r.Indeed, after the spin update on vertex v from m to r, the spin m on the left side of v, say ṽ, has two nearest neighbors with spin m and two nearest neighbors with spin r and the path flips it to r at zero energy cost.Hence, after this step the spin on the left side of ṽ has two nearest neighbors with spin m and two nearest neighbors with spin r and the path flips it from m to r without changing the energy.This construction is repeated until the last step, when the path flips the last spin m to r.Indeed, in this case the spin m that is flipping has two nearest neighbors with spin r, one nearest neighbor m and the fourth nearest neighbor has spin different from r and m.Thus, after this last flip the energy is reduced by 1 and the claim is proved.Let us now consider the cluster depicted in Figure 6(b).In this case, we construct a downhill path given by two steps.First, it flips from m to r the spin m on the vertex v and the energy does not change.Then, it flips from m to r a spin m, say on vertex v, which, after the previous flip, has three nearest neighbors with spin r and one nearest neighbor m.Thus, by this flip the energy is reduced by two and the claim is verified.In view of above, we conclude that any configuration σ ∈ M pos ∪ Mpos does not contain any m-rectangle in a sea of spin r or with at least two consecutive sides adjacent to clusters of spin r for any m, r ∈ S, see Figure 6(a).Furthermore, for any m, r ∈ S, r = m, we get that in σ there is not a spin m that has two nearest neighbors with spin m inside a rectangle with minimum side of length one and two nearest neighbors with spin r belonging to two different m-rectangles, see Figure 6(b).
Step 4. Let us now consider the case in which σ have stable tiles as in Figure 5(a)-(q).First, we note that the stable v-tiles as in Figure 5(i), (m) and (q) are such that any spinupdate on vertex v strictly increases the energy by at least one.Thus, every σ ∈ M pos ∪ Mpos may have strips of thickness one as long as they are adjacent to two strips with different spins.

(c)
Figure 7: Illustration of an m-rectangle, that we color black, adjacent to two r-rectangles, that we color light gray.Furthermore, we color gray those t-rectangles with t ∈ S\{r, m}.Now note that in view of the stable v-tiles depicted in Figure 5(h), (l) and (p), we have that for any m, r ∈ S, m = r, in σ an m-rectangle may interact with an r-rectangle.More precisely, either an m-rectangle has a side adjacent to a side of an r-rectangle or there exists a vertex with spin s ∈ S\{m, r} that has two nearest neighbors with spin s, a nearest neighbor m inside the m-rectangle and a nearest neighbor r belonging to the r-rectangle.Now note that in the previous step we studied the case in which an m-rectangle has two consecutive sides adjacent two r-rectangles, for any m, r ∈ S\{1}, with the sides on the interfaces with the same length, see Figure 6(a).By taking into account also the stable tiles depicted in Figure 5(h), (l) and (p), now σ may have an m-rectangle which has two consecutive sides adjacent to two r-rectangles but the sides on the interfaces have not the same length, see Figure 6 where the black rectangle denotes the m-rectangle.It is easy to state that any configuration σ ∈ M pos ∪ Mpos has not the clusters depicted in Figure 7(a) and (b).Indeed, following the same strategy as in Step 3.3 above, we construct a path that flips from m to r all the spins m adjacent to the side of length ℓ such that along this path the energy decreases.Hence, let us consider the case depicted in Figure 7(c).We prove that if σ has this type of cluster surrounded by stable tiles, then σ ∈ Mpos .Indeed, in this case σ does not communicate with a configuration with energy strictly lower but there exists a path which connects σ and other configurations with the same energy.Indeed, we define a path which flips from m to r the spin m adjacent to an r-rectangle.In particular, at any step the spin m that is flipping has two nearest neighbors with spin m and two nearest neighbors with spin r, see for instance the path depicted in Figure 8 where the black rectangle denotes the m-rectangle.Since the sequence of configurations in Figure 8 have the same energy value and they are connected with each other by means a path, they belong to a stable plateau.In particular, the energy along the path is constant.) such that H pos (ω i ) = H pos (ω j ), for any i, j = 0, . . ., n.Since all the configurations depicted have the same energy value and they are connected by means a path, they belong to a stable plateau.
Step 5. Let us now consider the case in which σ may have stable tiles as in Figure 5(a)-(r).Let us focus on the stable tile depicted in Figure 5(r).Note that we have only to study the case in which the three nearest neighbors of v with spins r, s, t ∈ S\{m} are different from each other.Indeed, all the other cases are not stable tiles for σ by item (1) of Lemma 5.7.To aid the reader we refer to Figure 9 for a pictorial illustration of the tile depicted in Figure 5(r), where we represent r, m, s, t respectively by , , , and where we assume that r, s, t ∈ S\{1, m}.Assume that this type of tile belongs to the configuration σ.
Figure 9: Example of a v-tile equal to the one depicted in Figure 5(r).We do not color the vertices w 1 and w 2 since in the proof they assume different value in different steps.
Step 5.1.If n m (v 1 ) = 4, then both v 2 and v 4 must have two nearest neighbors with their spin value and two nearest neighobors m, i.e., n m (v i ) = 2 and n σ(vi) (v i ) = 2 for i = 2, 4. Otherwise, the v 2 -tile and the v 4 -tile would be not stable for σ by item (1) of Lemma 5.7 and we decrease the energy by a single spin update.It follows that the v 2 -tile and v 4 -tile coincide with those depicted in Figure 5(f), thus we are in the same situation of Step 3 and Step 4 and we use the same argument.
Step 5.2.If n m (v 1 ) = 3, similarly to the previous step, at least one between v 2 and v 4 have two nearest neighbors with its same spin value and two nearest neighbors of spin m.Hence, we are in an analogous situation as in Step 3 and in Step 4 and we conclude that such a tile belongs to a configuration which is either unstable for σ or it belongs to a stable plateau.
Step 5.3.Let us now consider the case in which v 1 has only a nearest neighbor with spin m, i.e., the spin m on vertex v.We anticipate this case because it will be useful to study the case n m (v 1 ) = 2 in the next step.Let r ′ be a spin value such that n r ′ (v 1 ) ≥ 1.Along the path ω := (σ, σ v,r , (σ v,r ) v1,r ′ ) the energy decreases.Indeed, we have Note that the first equality follows by the fact that σ v,r is obtained by σ by flipping the spin on vertex v from m to r and m, r ∈ S\{1}.On the other hand, the second equality follows by the fact that (σ v,r ) v1,r ′ is defined by σ v,r by flipping the spin on vertex v 1 from m to r ′ and v 1 has not any nearest neighbor with spin m in σ v,r .Thus, the energy decreases by the number of the nearest neighbor with spin r ′ of v 1 in σ v,r and also by h when r ′ = 1, see (2.8).It follows that in this case the tiles as depicted in Figure 5(r) do not belong to any configuration σ ∈ M pos ∪ Mpos .
Step 5.4.The last case that we have to study is the one in which v 1 has two nearest neighbors with spin m.Obviously one of these is the vertex v.If the other spin m lies either on the vertex w 1 or on the vertex w 2 , then, it is in a configuration that falls in Step 5.1.Hence, we conclude and in the sequel we assume that v 1 has the second nearest neighbor m on the vertex which belongs to the same row where there are v 1 and v.If v 1 has at least one nearest neighbor with a spin among r, s, t, say r, then along the path ω = (σ, σ v,r , (σ v,r ) v1,r ) the energy decreases.Indeed, we have (5.25) In particular, the second inequality follows by (2.8), by r = 1 and by the fact that v 1 has at least one nearest neighbor with spin r in σ.Thus in this case a tile as depicted in Figure 5(r) does not belong to any configuration σ ∈ M pos ∪ Mpos .Hence assume that v 1 has two nearest neighbors with spin m on vertices v and v 5 and that for any r ′ / ∈ {r, t, z}, n r ′ (v 1 ) ∈ {1, 2}.If n r ′ (v 1 ) = 2, then we construct a path ω := (σ, σ v,r , (σ v,r ) v1,r ′ ) along which the energy decreases.Indeed, we have where the first equality holds because m and r are different from 1. On the other hand, the second equality follows by (2.8) the fact that v 1 has always two nearest neighbors with spin r ′ in σ v,r since r = r ′ .Otherwise, n r ′ (v 1 ) = 1, see Figure 10(a) where we represent r ′ by .Without changing the energy, we flip from m to a spin among r, s, t, say r, the spin on vertex v, see Figure 10(b).We may repeat the procedure of above discussion by replacing v with v 1 and v 1 with v 5 .In any case, we conclude that a v-tile as in Figure 5(r) does not belong to any configuration σ ∈ M pos ∪ Mpos .But at a certain point, we surely find a vertex u such that the u-tile is analogous to the one centered in the vertex v in Step 5.3, see Figure 10(c).Note that by periodic boundary conditions this vertex may be the one on the right side of the vertex v 3 , thus we conclude the proof of Case 5. Additionally, we note that in view of the above construction we have that the stable v-tiles as in Figure 5(i) belong only to a configuration σ ∈ M pos ∪ Mpos when they belong to a strip, i.e., when v lies either on a row or on a column in which all the spins have the same value.
Step 6.Finally, we consider the case in which σ has any possible stable tiles depicted in Figure 5(a)-(s).Note that the v-tile depicted in Figure 5(s) is such that any spin-update on vertex v strictly increases the energy by at least h.Hence, using these stable tiles together with those as in Figure 5(o) and (q), we get that a configuration σ ∈ M pos ∪ Mpos may have a cluster as depicted in Figure 11, i.e., σ may have a 1-rectangle with a side of length one and the other one of length larger than or equal to two.This rectangle staisfies the following conditions: there are no two consecutive sides adjacent to rectangles of the same spins and the sides of length different from one have to be adjacent to rectangles with spins different from each other.
Hence, we conclude that if σ is characterized by the stable tiles depicted in Figure 5 We are now able to prove Proposition 4.2.
Proof of Proposition 4.2.Let M pos := (M pos \{1, . . ., q}) ∪ Mpos .In order to conclude the proof, it is enough to focus on the configurations belonging to M pos .Given η ∈ M pos , we prove that V pos η is smaller than or equal to V * := 2 < Γ pos (m, X s pos ) for any m ∈ X m pos .
Let us first give an outline of the proof.First, we estimate the stability level for those configurations in M 2 pos ⊂ M pos that are characterized by two or more strips of different spins, see Figure 12(a)-(b).Second, we compute the stability level for those configurations in M 3 pos ⊂ M pos that are characterized by a sea of m, for any m ∈ S\{1}, with some rectangles of spins 1 which do not interact among each other, see Figure 12(c).Third, we consider the case of those configurations in M 4 pos ∪ Mpos ⊂ M pos that are covered by interacting rectangles, see Figure 12(d).Note that any configuration in M pos belongs to one of the three cases above.Indeed, M pos = M 2 pos ∪ M 3 pos ∪ M 4 pos ∪ Mpos and, any η ∈ M pos is such that it has at least two strips, or at least an isolated 1-rectangle inside either a sea of spins m or a strip of spins m, or a couple of interacting rectangles.See Figure 12(e).Case 1.Let us begin by assuming that η only contains either horizontal or vertical strips.For concreteness, consider the case represented in Figure 12(a).The case represented in Figure 12(b) may be studied similarly.Assume that η has an m-strip a×K adjacent to an r-strip b×K with a, b ∈ Z, a, b ≥ 1. Assume that m, r ∈ S, m = 1.Let η be the configuration obtained from η by flipping all the spins m belonging to the m-strip from m to r.We define a path ω : η → η as the concatenation of a paths ω (1) , . . ., ω (a) .In particular, for any i = 1, . . ., a − 1, where η 0 ≡ η and η i is the configuration in which the initial m-strip is reduced to a strip (a − i) × K and the r-strip to a strip (b + i) × K.For any i = 1, . . ., a − 1, the path ω (i) flips consecutively from m to r those spins m belonging to the column next to the r-strip.In particular, note that for any i = 1, . . ., a − 1, ω (i) 0 , v is the center of a tile as in Figure 5(c).Thus, using (2.8), we have (5.27) Otherwise, if i = 2, . . ., K, then v i has one nearest neighbor with spin m and at most three nearest neighbors with spin r.Thus, using (2.8), we have In view of (5.27)-(5.31),we get H(η) > H(η).Furthermore, since the maximum energy value is reached at the first step, we get Case 2. For any m ∈ S, m = 1, let us now consider the local minimum η characterized by a sea of spins m with some not interacting 1-rectangles, see for instance Figure 12(c).We distinguish two cases: (i) η has at least a rectangle R ℓ1×ℓ2 of spins 1 with its minimum side of length ℓ := min{ℓ 1 , ℓ 2 } larger than or equal to ℓ * , (ii) η has only rectangles R ℓ1×ℓ2 of spins 1 with a side of length ℓ smaller than ℓ * .
Case 3. Finally, we focus on a local minimum η covered by adjacent rectangles, see for instance Figure 12(d).Let m, r ∈ S\{1}, m = r.Indeed, any m-rectangle has to interact with at least an r-rectangle with r = 1 since otherwise it could be surrounded by spins 1 that is in contradiction with η ∈ M pos by Proposition 5.3.Assume that η has an m-rectangle R := R a×b and an r-rectangle R := R c×d such that the m-rectangle R has a side of length a adjacent to a side of the r-rectangle R of length c ≥ a.The case c < a may be studied by interchanging the role of the spins m and r.Given η the configuration obtained from η by flipping from m to r all the spins m belonging to R, we construct a path ω : η → η as the concatenation of b paths ω (1) , . . ., ω (b) .In particular, for any i = 1, . . ., b − 1, ω (i) := (ω , where η 0 ≡ η and η i is the configuration in which the initial r-rectangle R is reduced to a rectangle c × d with a protuberance a × i and the initial m-rectangle R is reduced to a rectangle a × (b − i).For any i = 1, . . ., b − 1, the path ω (i) flips consecutively from m to r those spins m adjacent to the side of length a of the m-rectangle a × (b − i).Given v the vertex in which ω 0 the vertex v has two nearest neighbors with spin m, one nearest neighbor with spin r and one nearest neighbor with spin different from m, r.Thus, using (2.8), we get (5.37) j−1 the vertex v has two nearest neighbors m and two nearest neighbors r, then (5.38) a−1 the vertex v has one nearest neighbor with spin m, two nearest neighbors with spin r and one nearest neighbor with spin different from m, r.Thus, according to (2.8) we get (5.39) Then, for any i = 1, . . ., b − 1, the maximum energy value along ω (i) is reached at the first step.Finally, we define a path ω (b) := (ω ℓ = η) that flips consecutively from m to r the spins m belonging to the remaining m-rectangle a × 1.In particular, if v i is the vertex whose spin is flipping in ω (b) i−1 at step i, if i = 1, then v 1 has two nearest neighbors with spin m and at most two nearest neighbors with spin r, i.e., (5.40) Otherwise, if i = 2, . . ., a, then v i has one nearest neighbor with spin m and at most three nearest neighbors with spin r.Thus, using (2.8), we have Thanks to (5.37)-(5.41)and since by Proposition 5.3 we have a ≥ 2, we get H(η) > H(η).Moreover, since the maximum along ω is reached at the first step, by (5.37) we get

Minimal gates and tube of typical trajectories
In this section we investigate on the minimal gates and the tube of typical paths for the transition from any m ∈ X m pos to X s pos = {1}.We further identify the union of all minimal gates also for the transition from a metastable state to the other metastable states.

Identification of critical configurations for the transition from a metastable to the stable state
The goal of this subsection is to investigate the set of critical configurations for the transition from any m ∈ X m pos to X s pos = {1}.The idea of the proof of the following lemmas and proposition generalizes the proof of similiar results given in [24,Section 6] for the Blume Capel   A two dimensional polyomino on Z 2 is a finite union of unit squares.The area of a polyomino is the number of its unit squares, while its perimeter is the cardinality of its boundary, namely, the number of unit edges of the dual lattice which intersect only one of the unit squares of the polyomino itself.Thus, the perimeter is the number of interfaces on Z 2 between the sites inside the polyomino and those outside.The polyominoes with minimal perimeter among those with the same area are said to be minimal polyominoes.Proof.Let m ∈ {2, . . ., q}.From the definition of the Hamiltonian H pos , (2.2), we get that the presence of disagreeing edges increases the energy, thus in order to identify the bottom of D m pos we have to consider those configurations σ ∈ D m pos in which the ℓ * (ℓ * −1)+1 spins different from m belong to a single cluster.Moreover, given the number of the disagreeing edges, the presence of each spin 1 decreases the energy by h compared of the presence with other spins.Hence, the single cluster is full of spins 1, say C 1 (σ), and it is inside a homogenous sea of spins m.Arguing like in the second part of the proof of Proposition 5.2, we have that 4ℓ * is the minimal perimeter of a polyomino of area ℓ * (ℓ * − 1) + 1.Thus, for any σ ∈ F (D m pos ), C 1 (σ) must have perimeter 4ℓ * .Hence, all the characteristics given in (6.4) are verified.Let us now prove (6.5).By (4.7) we get that W pos (m, X s pos ) ⊂ F (D m pos ), that is H pos (F (D m pos )) = H pos (W pos (m, X s pos )).Thus, (6.5) is satisfied since for any η ∈ W pos (m, X s pos ), In the next corollary we show that every optimal path from m ∈ X m pos to X s pos = {1} visits at least once F (D m pos ), i.e., we prove that F (D m pos ) is a gate for the transition from m to X s pos .Proof.Every path from m ∈ X m pos to the stable configuration 1 has to pass through the set V m k := {σ ∈ X : N m (σ) = k} for any k = |V |, . . ., 0. In particular, given k * := ℓ * (ℓ * − 1) + 1, any ω = (ω 0 , . . ., ω n ) ∈ Ω opt m,X s pos visits at least once the set V m |Λ|−k * ≡ D m pos .Hence, there exists i ∈ {0, . . .n} such that ω i ∈ D m pos .Since from (6.5) we have that the energy value of any configuration belonging to F (D m pos ) is equal to the min-max reached by any optimal path from m to X s pos , we conclude that ω i ∈ F (D m pos ).
In the last result of this subsection, we prove that, for any m ∈ X m pos , every optimal path ω ∈ Ω opt m,X s pos is such that ω ∩ W pos (m, X s pos ) = ∅.Hence, we show that W pos (m, X s pos ) is a gate for the transition m → X s pos .
Proposition 6.1.If the external magnetic field is positive, then for any m ∈ X m pos each path ω ∈ Ω opt m,X s pos visits W pos (m, X s pos ).Hence, W pos (m, X s pos ) is a gate for the transition m → X s pos .
Proof.For any m ∈ S, m = 1, let Dm pos and Dm pos be the subsets of F (D m pos ) defined as follows.Dm pos is the set of those configurations of F (D m pos ) in which the boundary of the polyomino C 1 (σ) intersects each side of the boundary of its smallest surrounding rectangle R(C 1 (σ)) on a set of the dual lattice Z 2 + (1/2, 1/2) made by at least two consecutive unit segments, see Figure 14(a).On the other hand, Dm pos is the set of those configurations of F (D m pos ) in which the boundary of the polyomino C 1 (σ) intersects at least one side of the boundary of R(C 1 (σ)) in a single unit segment, see Figure 14(b) and (c).In particular note that F (D m pos ) = Dm pos ∪ Dm pos .The proof proceeds in five steps.
Step 1.Our first aim is to prove that From (4.7) we have W pos (m, X s pos ) ∪ W ′ pos (m, X s pos ) ⊆ Dm pos .Thus we reduce our proof to show that σ ∈ Dm pos implies σ ∈ W pos (m, X s pos ) ∪ W ′ pos (m, X s pos ).Note that this implication is not straightforward, since given σ ∈ Dm pos , the boundary of the polyomino C 1 (σ) could intersect the other three sides of the boundary of its smallest surrounding rectangle R(C 1 (σ)) in a proper subsets of the sides itself, see Figure 14(d) for an illustration of this hypothetical case.Hence, consider σ ∈ Dm pos and let R(C In view of the proof of Lemma 6.1 we have that C 1 (σ) is a minimal polyomino and by [24,Lemma 6.16] it is also convex and monotone, i.e., its perimeter of value 4ℓ * is equal to the one of R(C 1 (σ)).Hence, the following equality holds In particular, (A.10) is satisfied only by a = −b.Now, let R be the smallest rectangle surrounding the polyomino, say C1 (σ), obtained by removing the unit protuberance from C 1 (σ).
If C 1 (σ) has the unit protuberance adjacent to a side of length ℓ * + a, then R is a rectangle (ℓ * + a) × (ℓ * − a − 1).Note that R must have an area larger than or equal to the number of spins 1 of the polyomino C1 (σ), that is ℓ * (ℓ * − 1).Thus, we have Since a ∈ Z, −a 2 + a ≥ 0 is satisfied only if either a = 0 or a = 1.In both cases we get that R is a rectangle of side lengths ℓ * and ℓ * − 1.Thus, if the protuberance is attached to one of the longest sides of R, then σ ∈ W pos (m, X s pos ), otherwise σ ∈ W ′ pos (m, X s pos ).In any case we conclude that (A.9) is satisfied.
Step 2. For any m ∈ X m pos and for any path ω = (ω 0 , . . ., We claim that g m (ω) = ∅.Let ω = (ω 0 , . . ., ω n ) ∈ Ω opt m,X s pos and let j * ≤ n be the smallest integer such that after j * the path leaves D m,+ pos , i.e., (ω j * , . . ., ω n ) ∩ D m,+ pos = ∅.Since ω j * −1 is the last configuration in D m,+ pos , it follows that ω j * ∈ D m pos and, by the proof of Corollary 6.1, we have that is not feasible since ω j * and ω j * −1 differ by a single spin update which increases the number of spins 1 of at most one.Then, j * ∈ g m (ω) and the claim is proved.
Step 3. We claim that for any path ω ∈ Ω opt m,X s pos one has ω i ∈ Dm pos for any i ∈ g m (ω).We argue by contradiction.Assume that there exists i ∈ g m (ω) such that ω i / ∈ Dm pos and ω i ∈ Dm pos .Since ω i−1 is obtained from ω i by flipping a spin 1 to m and since any configuration belonging to Dm pos has all the spins 1 with at least two nearest neighbors with spin 1, using (2.8) we have In particular, from (6.12) we get a contradiction.Indeed, where the first equality follows by (6.5).Thus by (6.13) ω is not an optimal path, which is a contradiction, the claim is proved and we conclude the proof of Step 3.
Step 4. Now we claim that for any m ∈ X m pos and for any path ω ∈ Ω opt m,X s pos , Using Corollary 6.1, for any m ∈ X m pos and any path ω ∈ Ω opt m,X s pos there exists an integer i such that ω i ∈ F (D m pos ).Assume by contradiction that ω i+1 ∈ D m pos .In particular, since ω i and ω i+1 have the same number of spins m, note that ω i+1 is obtained by flipping a spin 1 from 1 to t = 1.Since ω i (v) = t for every v ∈ V , the above flip increases the energy, i.e., H pos (ω i+1 ) > H pos (ω i ).Hence, using this inequality and (6.5), we have which implies the contradiction because ω is not optimal.Thus ω i+1 / ∈ D m pos and similarly we show that also ω i−1 / ∈ D m pos .
Step 5.In this last step of the proof we claim that for any m ∈ X m pos and for any path ω ∈ Ω opt m,X s pos there exists a positive integer i such that ω i ∈ W pos (m, X s pos ).Arguing by contradiction, assume that there exists ω ∈ Ω opt m,X s pos such that ω ∩ W pos (m, X s pos ) = ∅.Thanks to Corollary 6.1, we know that ω visits F (D m pos ) and thanks to Step 4 we have that the configurations along ω belonging to F (D m pos ) are not consecutive.More precisely, they are linked by a sub-path that belongs either to D m,+ pos or D m,− pos .If n is the length of ω, then let j ≤ n be the smallest integer such that ω j ∈ F (D m pos ) and such that (ω j , . . ., ω n ) ∩ D m,+ pos = ∅, thus, j ∈ g m (ω) since j plays the same role of j * in the proof of Step 2. Using (A.9), Step 3 and the assumption ω ∩ W pos (m, X s pos ) = ∅, it follows that ω j ∈ W ′ pos (m, X s pos ).Moreover, starting from ω j ∈ F (D m pos ) the energy along the path decreases only by either (i) flipping the spin in the unit protuberance from 1 to m, or (ii) flipping a spin, with two nearest neighbors with spin 1, from m to 1.
Since by the definition of j we have that ω j−1 is the last that visits D m,+ pos , ω j+1 / ∈ D m,+ pos , (i) is not feasible.Considering (ii), we have H pos (ω j+1 ) = H pos (m) + Γ pos (m, X s pos ) − h.Starting from ω j+1 we consider only moves which imply either a decrease of energy or an increase by at most h.Since C 1 (ω j+1 ) is a polyomino ℓ * × (ℓ * − 1) with a bar made of two adjacent unit squares on a shortest side, the only feasible moves are (iii) flipping a spin, with two nearest neighbors with spin m, from m to 1, (iv) flipping a spin, with two nearest neighbors with spin 1, from 1 to m.
By means of the moves (iii) and (iv), the process reaches a configuration σ in which all the spins are equal to m except those, that are 1, in a connected polyomino C 1 (σ) that is convex and such that R(C 1 (σ)) = R (ℓ * +1)×(ℓ * −1) .We cannot repeat the move (iv) otherwise we get a configuration that does not belong to D m pos .While applying one time (iv) and iteratively (iii), until we fill the rectangle R (ℓ * +1)×(ℓ * −1) with spins 1, we get a set of configurations in which the one with the smallest energy is σ such that C 1 (σ) ≡ R(C 1 (σ)).Moreover, from any configuration in this set, a possible move is reached by flipping from m to 1 a spin m with three nearest neighbors with spin m that implies to enlarge the circumscribed rectangle.This spin-flip increases the energy by 2 − h.Thus, we obtain which is a contradiction by the definition of an optimal path.Note that the last inequality follows by 2 > h(ℓ * − 1) since 0 < h < 1, see Assumption 4.1.It follows that it is not possible to have ω ∩ W pos (m, X s pos ) = ∅ for any ω ∈ Ω opt m,X s pos , namely W pos (m, X s pos ) is a gate for this type of transition.

Minimal gates for the transition from a metastable state to the other metastable states
This subsection is devoted to the study of the transition from a metastable state to the set of the other metastable configurations.In Propositions 6.3 and 6.6 we identify geometrically two gates for this type of transition and in Theorem 4.6 we show that the union of these sets gives the union of all the minimal gates for the same transition.Furthermore, in this subsection we also give some more details for the transition from any metastable state to the stable configuration 1.More precisely, in Proposition 6.5 we prove that for any m ∈ X m pos almost surely any optimal path ω ∈ Ω opt m,X s pos does not visit any metastable state different from the initial one during the transition.Let us begin by proving the following useful lemma.Lemma 6.2.For any m ∈ {2, . . ., q}, let η ∈ B1 ℓ * −1,ℓ * (m, 1) and let η ∈ X a configuration which communicates with η by one step of the dynamics.If the external magnetic field is positive, then either H pos (η) < H pos (η) or H pos (η) > H pos (η).
Proof.Since η and η differ by a single-spin update, let us define η := η v,t for some v ∈ V and t ∈ S, t = η(v).Note that η ∈ B1 ℓ * −1,ℓ * (m, 1) implies that η is characterized by all spins m except those that are 1 in a quasi-square (ℓ * − 1) × ℓ * with a unit protuberance on one of the longest sides.In particular, for any v ∈ V , either η(v) = m or η(v) = 1.If η(v) = m, then for any t ∈ S\{m}, depending on the distance between the vertex v and the 1-cluster, we have (6.17) Otherwise, if η(v) = 1, for any t ∈ S\{1}, depending on the distance between the vertex v and the boundary of the 1-cluster, we get We conclude that H pos (η) = H pos (η).
Exploiting the equality Φ pos (m, X s pos ) = Φ pos (m, X m pos \{m}) for any m ∈ X m pos , we are now able to state the following corollary and proposition.) is a gate for the transition m → X m pos \{m}.Proof.Thanks to (6.19) the proof is analogous to the one of Corollary 6.1.We refer to Appendix A.2.1 for the explicit proof .Proposition 6.3.If the external magnetic field is positive, for any m ∈ X m pos , W pos (m, X s pos ) is a gate for the transition m → X m pos \{m}.Proof.Thanks to (6.19) the proof is analogous to the one of Proposition 6.1.See Appendix A.2.2 for the detailed proof.Given m ∈ X m pos , the reader may be surprised that W pos (m, X s pos ) is a minimal gate for both the transitions m → X s pos and m → X m pos \{m}.Intuitively, the set Ω m,X m pos \{m} is partitioned in two non-empty subsets, i.e., the set containing those paths ω ∈ Ω m,X m pos \{m} such that ω ∩ C 1 More precisely, in Proposition 6.5 we show that almost surely the process started in m ∈ X m pos does not visit any other metastable states before hitting the stable configuration X s pos = {1}.In order to prove this result, first we need to introduce the following habitat and to show that almost surely during the transition from a metastable to the stable state the process does not exit from it.For any m ∈ X m pos , let where δ is the minimum energy gap between an optimal and a non-optimal path from m to X s pos .Note that A pos is a cycle and that the choice to give some results on the dynamics from a metastable to the stable state inside A pos is justified by the following result.Proof.By Proposition 4.1 we have X s pos = {1} and by the definition (6.25) we have 1 ∈ A pos .Hence, F (A pos ) = X s pos = {1}.Furthermore, by (6.25) we also get that X m pos ⊂ A pos .Hence, using Theorem 4.1 we have that V (A pos ) = Γ m pos .Finally, (6.26) is verified thanks to Equation (2.20) of [49,Theorem 2.17] applied to the cycle A pos .We are now able to prove the following result.Proposition 6.5 (Study of the transition from any m ∈ X m pos to X s pos ).If the external magnetic field is positive, then for any m ∈ X m pos we have that every optimal path from m to X s pos almost surely does not intersect other metastable states.More precisely, (6.27) pos and, for some j < n, let ω j ∈ W pos (m, X s pos ).By Corollary 4.2 and by Corollary 4.3 we get that almost surely the process started in m ∈ X m pos visits W pos (m, X s pos ) before hitting X s pos ∪ X m pos \{m}.Hence, almost surely we have (ω 0 , . . ., ω j ) ∩ (X s pos ∪ X m pos \{m}) = ∅.(6.28) Thus, our claim is to show that starting from ω j , the process arrives in X s pos before visiting X m pos \{m}.By Lemma 6.2, we have that ω j+1 does not have the same energy value of ω j .Thus, starting from ω j , the path passes to a configuration with energy strictly lower or strictly higher than H pos (ω j ).More precisely, for some v ∈ V and some t ∈ S, let ω j+1 := ω v,t j .We have to consider the following possibilities: passes through η and such that it reaches its maximum energy only in this configuration.In particular, the optimal path is defined by modifying the reference path ω of Definition 5.1 in a such a way that ωℓ * (ℓ * −1)+1 = η in which C 1 (η) is a quasi-square ℓ * × (ℓ * − 1) with a unit protuberance.This is possible by choosing the intial vertex (i, j) such that during the construction the cluster C 1 (ω ℓ * (ℓ * −1) ) coincides with the quasi-square in η and in the next step the unit protuberance is added in the site as in η.It follows that ω ∩ W pos (m, X s pos ) = {η} and by the proof of Lemma 5.5 we get arg max ω H pos = {η}.To conclude, we prove (4.8), i.e., that W pos (m, X s pos ) is the unique minimal gate for the transition m → X s pos .Note that the above reference paths ω reach the energy Φ pos (m, X s pos ) only in W pos (m, X s pos ).Thus, we get that for any η 1 ∈ W pos (m, X s pos ), the set W pos (m, X s pos )\{η 1 } is not a gate for the transition m → X s pos since, in view of the above construction, we have that there exists an optimal path ω such that ω ∩ W pos (m, X s pos )\{η 1 } = ∅.Note that the uniqueness of the minimal gate follows by the condition 2  h / ∈ N, see Assumption 4.1(ii).
Remark 6.1.A saddle η ∈ S(σ, σ ′ ) is unessential if for any ω ∈ Ω opt σ,σ ′ such that ω ∩ η = ∅ the following conditions are both satisfied: (i) {argmax ω H}\{η} = ∅, (ii) there exists ω ′ ∈ Ω opt σ,σ ′ such that {argmax ω ′ H} ⊆ {argmax ω H}\{η}.Proof of Theorem 4.6 By Proposition 6.3 we have that the set given in (a) is a gate for the transition m → X m pos \{m}.Hence, our aim is to prove that W pos (m, X s pos ) is a minimal gate for the same transition.In order to show that this set satisfies the definition of minimal gate given in Subsection 3.1, we show that for any η ∈ W pos (m, X s pos ) there exists an optimal path ω ′ ∈ Ω opt m,X m pos \{m} such that ω ′ ∩ (W pos (m, X s pos )\{η}) = ∅.We construct this optimal path ω ′ as the reference path ω * defined in the proof of the upper bound of Proposition 6.2 in such a way that at the step k * − 1 the rectangular ℓ * × (ℓ * − 1) s-cluster is as in η without the protuberance.For k * ≤ k ≤ k * + ℓ * − 1, we proceed as follows.At step k * the unit protuberance is added in the same position as in η, and in the following steps the same side is filled flipping consecutively to s spins 1 that have two nearest neighbors with spin s.Thus, ω ′ ∩ W pos (m, X s pos ) = {η} and the condition of minimality is satisfied.By Proposition 6.6 the set depicted in (b) is a gate for the transition m → X m pos \{m}.Thus, our aim is to prove that z∈X m pos \{m} W pos (z, X s pos ) is a minimal gate for the same transition.Similarly to the previous case we show that for any η ∈ z∈X m pos \{m} W pos (z, X s pos ) there exists an optimal path ω ′ ∈ Ω opt m,X m pos \{m} such that ω ′ ∩ ( z∈X m pos \{m} W pos (z, X s pos )\{η}) = ∅.We define this optimal path ω ′ as the reference path ω * constructed in the proof of the upper bound of Proposition 6.2 in such a way that at the step k * − 1 the rectangular ℓ * × (ℓ * − 1) s-cluster is as in η without the protuberance.For k * ≤ k ≤ k * + ℓ * − 1, we proceed as follows.At step k * the unit protuberance is added in the same position as in η, and in the following steps the same side is filled flipping consecutively to s spins 1 that have two nearest neighbors with spin s.Thus, ω ′ ∩ z∈X m pos \{m} W pos (z, X s pos ) = {η} and the condition of minimality is verified.Thus is an unessential saddle for the transition m → X m pos \{m}.To this end we prove that any η as in (6.32) satisfies conditions Remark 6.1 (i) and (ii).Indeed, let ω ∈ Ω opt m,X m pos \{m} such that ω ∩ {η} = ∅.Note that condition (i) in Remark 6.1 is satisfied since ω intersects at least once both W pos (m, X s pos ) and z∈X m pos \{m} W pos (z, X s pos ).Next we define an optimal path ω ′ ∈ Ω opt m,X m pos \{m} in order to prove that also condition (ii) in Remark 6.1 is satisfied.From Propositions 6.3 and 6.6, there exist η * 1 ∈ ω ∩ W pos (m, X s pos ) and η * 2 ∈ ω ∩ z∈X m pos \{m} W pos (z, X s pos ).Thus, we construct ω ′ as the reference path defined in the proof of the upper bound of Proposition 6.2 in such a way that ω

Tube of typical paths: proof of the main results
In order to give the proof of Theorem 4.7, first we prove the following lemmas.
Thus, the smallest energy increase is given by h by flipping to m a spin 1 on a vertex v 1 centered in a tile as in Figure 5 ).In η 2 , for any v ∈ V the corresponding v-tile is one among those depicted in Figure 5(a), (b), (d), (e), (n) and (l) with r = 1.Since H pos (η 2 ) = H pos (η 1 ) + h, the spin-flips on a vertex whose tile is one among those depicted in Figure 5(a), (b), (d), (e) lead to H pos (σ 2 ) − H pos (η 1 ) ≥ 2. Thus, as in the previous case, the smallest energy increase is given by flipping to m a spin 1 on a vertex v 2 centered in a tile as Figure 5(n).Note that starting from η 2 the only spin-flip which decreases the energy leads to the bottom of C(η 1 ), namely in η 1 .Iterating the strategy, the same arguments hold as long as the uphill path towards F (∂C(η 1 )) visits η ℓ−1 ∈ B1 ℓ−1,ℓ−1 (m, 1).Indeed, in this type of configuration for any v ∈ V the corresponding v-tile is one among those depicted in Figure 5(a), (b), (d), (e), (n) and unstable tile (s) with t = r = s = m, and it is possible to decrease the energy by passing to a configuration that does not belong to C(η 1 ).More precisely, there exists a vertex w such that its tile is as the one in Figure 5   .Note that if v 1 is adjacent to a side of length ℓ 2 , then η 2 ∈ B1 ℓ1,ℓ2 (m, 1), otherwise η 2 ∈ B1 ℓ2,ℓ1 (m, 1).Without loss of generality, let us assume that η 2 ∈ B1 ℓ1,ℓ2 (m, 1).By simple algebraic calculation we obtain that In η 2 for any v ∈ V the corresponding v-tile is one among those depicted in Figure 5(a), (b), (d), (e), (n) and (l) with r = 1.By flipping to 1 a spin m on a vertex w whose tile is as the one depicted in Figure 5(l) with r = 1 the energy decreases by h and the process enters a cycle different from the previous one that is either the cycle C whose bottom is a local minimum belonging to Rℓ1+1,ℓ2 (m, 1), or a trivial cycle for which iterating this procedure the process enters C. Thus, B1 ℓ1,ℓ2 (m, 1) ⊆ ∂C(η 1 ).Similarly we prove that B1 ℓ2,ℓ1 (m, 1) ⊆ ∂C(η 1 ).Let us now note that starting from η 1 the smallest energy increase is h, and it is given by flipping to m a spin 1 on a vertex whose tile is as the one depicted in Figure 5(n) with r = m.Let us consider the uphill path ω started in η 1 and constructed by flipping to m all the spins 1 along a side of the rectangular ℓ 1 × ℓ 2 1-cluster, say one of length ℓ 1 .Using the discussion given in the proof of Lemma 6.3 and the construction of ω, we get that the process intersects ∂C(η) in a configuration σ belonging to B1 ℓ2−1,ℓ1 (m, 1).By simple algebraic computations, we obtain the following Since ℓ 2 ≥ ℓ * , it follows that H pos (σ) > H pos (η 2 ).Since by flipping to m (resp.1) the vertex centered in a tile as depicted in Figure 5(b), (e) (resp.(a)), the energy increase is largest than or equal to 2 + h, it follows that (6.40) is satisfied.
We are now able to prove Theorem 4.7 Proof of Theorem 4.7.Following the same approach as [58, Section 6.7], we geometrically characterize the tube of typical trajectories for the transition using the so-called "standard cascades".See [58,Figure 6.3] for an example of these objects.We describe the standard cascades in terms of the paths that are started in m and are vtj-connected to X s pos .See (4.16) for the formal definition and see [53,Lemma 3.12] for an equivalent characterization of these paths.We remark that any typical path from m to X s pos is also an optimal path for the same transition.In order to describe these typical paths we proceed similarly to [58,Section 7.4], where the authors apply the model-independent results given in Section 6.7 to identify the tube of typical paths in the context of the Ising model.Thus, we define a vtj-connected cycle-path that is the concatenation of both trivial and non-trivial cycles that satisfy (4.15).In Theorem 4.5 we give the geometric characterization of all the minimal gates for the transition m → X s pos .Let η 1 be a configuration belonging to one of these minimal gates.We begin by studying the first descent from η 1 both to m and to X s pos .Then, we complete the description of T X s pos (m) by joining the time reversal of the first descent from η 1 to m with the first descent from η 1 to X s pos .Let us begin by studying the standard cascades from η 1 to m.Since a spin-flip from 1 to t / ∈ {1, m} implies an increase of the energy value equal to the increase of the number of the disagreeing edges, we consider only the splin-flips from 1 to m on those vertices belonging to the 1-cluster.Thus, starting from η 1 and given v 1 a vertex such that η 1 (v 1 ) = 1, since H pos (η 1 ) = Φ pos (m, X s pos ), we get It follows that the only possibility in which the assumed optimality of the path is not contradicted is the one where n 1 (v 1 ) = 1 and n m (v 1 ) = 3.Thus, along the first descent from η 1 to m the process visits η 2 in which all the vertices have spin m except those, which are 1, in a rectangular cluster ℓ * × (ℓ * − 1), i.e., η 2 ∈ Rℓ * −1,ℓ * (m, 1).By Proposition 5.3, η 2 ∈ M 3 pos is a local minimum, thus according to (4.15) we have to describe its non-trivial cycle and its principal boundary.Starting from η 2 , the next configuration along a typical path is defined by flipping to m a spin 1 on a vertex v 2 on one of the four corners of the rectangular 1-cluster.Indeed, since H pos (η 2 ) = Φ pos (m, X s pos ) − 2 + h, we have and the only possibility in which the assumed optimality of the path is not contradicted is Then, a typical path towards m proceeds by eroding the ℓ * − 2 unit squares with spin 1 belonging to a side of length ℓ * − 1 that are corners of the 1-cluster and that belong to the same side of v 2 .Each of the first ℓ * − 3 spin-flips increases the energy by h, i.e., the smallest energy increase for any single step of the dynamics, and these uphill steps are necessary in order to exit from the cycle whose bottom is the local minimum η 2 .After these ℓ * − 3 steps, the process hits the bottom of the boundary of this cycle in a configuration η ℓ * ∈ B1 ℓ * −1,ℓ * −1 (m, 1), see Lemma 6.3.The last spin-update, that flips from 1 to m the spin 1 on the unit protuberance of the 1-cluster, decreases the energy by 2 − h.Thus, the typical path arrives in a local minimum η ℓ * +1 ∈ Rℓ * −1,ℓ * −1 (m, 1), i.e., it enters a new cycle whose bottom is a configuration in which all the vertices have spin 1, except those, which are 1, in a square (ℓ * − 1) × (ℓ * − 1) 1-cluster.Summarizing the construction above, we have the following sequence of vtj-connected cycles Iterating this argument, we obtain that the first descent from η 1 ∈ W pos (m, X s pos ) to m is characterized by the concatenation of those vtj-connected cycle-subpaths between the cycles whose bottom is the local minima in which all the vertices have spin equal to m, except those, which are 1, in either a quasi-square (ℓ − 1) × ℓ or a square (ℓ − 1) × (ℓ − 1) for any ℓ = ℓ * , . . ., 1, and whose depth is given by h(ℓ − 2).More precisely, from a quasi-square to a square, a typical path proceeds by flipping to m those spins 1 on one of the shortest sides of the 1-cluster.On the other hand, from a square to a quasi-square, it proceeds by flipping to m those spins 1 belonging to one of the four sides of the square.Thus, a standard cascade from η 1 to m is characterized by the sequence of those configurations that belong to Let us now consider the first descent from η 1 ∈ B1 ℓ * −1,ℓ * (m, 1) to X s pos = {1}.In order to not contradict the definition of an optimal path, we have only to consider those steps which flip to 1 a spin m.Indeed, adding a spin different from m and 1 leads to a configuration with energy value strictly larger than Φ pos (m, X s pos ).Thus, let w 1 be a vertex such that η 1 (w 1 ) = m.Flipping the spin m on the vertex w 1 , we get H pos (η w1,1 1 ) = Φ pos (m, X s pos ) + n m (w 1 ) − n 1 (w 1 ) − h, (6.47) and the only feasible choice is n m (w 1 ) = 2 and n 1 (w 1 ) = 2 in η 1 .Thus, η w1,1 1 ∈ B2 ℓ * −1,ℓ * (m, 1), namely the bar is now of length two.Arguing similarly, we get that along the descent to 1 a typical path proceeds by flipping from m to 1 the spins m with two nearest-neighbors with spin 1 and two nearest-neighbors with spin m belonging to the incomplete side of the 1-cluster.More precisely, it proceeds downhill visiting ηi ∈ Bi ℓ * −1,ℓ * (m, 1) for any i = 2, . . ., ℓ * − 1 and ηℓ * ∈ Rℓ * ,ℓ * (m, 1), that is a local minimum by Proposition 5.3.In order to exit from the cycle whose bottom is ηℓ * , the process crosses the bottom of its boundary by creating a unit protuberance of spin 1 adjacent to one of the four edges of the 1-square, i.e., visits {η ℓ * +1 } where ηℓ * +1 ∈ B1 ℓ * ,ℓ * (m, 1).Indeed, starting from ηℓ * ∈ Rℓ * ,ℓ * (m, 1) the energy minimum increase is obtained by flipping a spin m with three nearest-neighbors with spin 1 and one nearest-neighbor with spin m.Starting from {η ℓ * +1 }, a typical path towards 1 proceeds by enlarging the protuberance to a bar of length two to ℓ * − 1, thus it visits ηℓ * +i ∈ Bi ℓ * ,ℓ * (m, 1) for any i = 2, . . ., ℓ * − 1.Each of these steps decreases the energy by h, and after them the descent arrives in the bottom of the cycle, i.e., in the local minimum η2ℓ * ∈ Rℓ * ,ℓ * +1 (m, 1).Then, the process exits from this cycle through the bottom of its boundary, i.e., by adding a unit protuberance of spin 1 on any one of the four edges of the rectangular ℓ * × (ℓ * + 1) 1-cluster in η2ℓ * , see Lemma 6.4.Thus, it visits the trivial cycle {η 2ℓ * +1 }, where η2ℓ * +1 ∈ B1 ℓ * ,ℓ * +1 (m, 1) ∪ B1 ℓ * +1,ℓ * (m, 1).Note that the resulting standard cascade is different from the one towards m.Thus, summarizing the construction above, we have defined the following sequence of vtj-connected cycles Note that if η2ℓ * ∈ B1 ℓ * ,ℓ * +1 (m, 1), then the process enters the cycle whose bottom is a configuration belonging to Rℓ * +1,ℓ * +1 (m, 1).On the other hand, if η2ℓ * ∈ B1 ℓ * +1,ℓ * (m, 1), then the standard cascade enters the cycle whose bottom is a configuration belonging to Rℓ * ,ℓ * +2 (m, 1).In the first case the cycle has depth hℓ * , in the second case the cycle has depth h(ℓ * − 1).Iterating this argument, we get that the first descent from η 1 to X s pos = {1} is characterized by vtj-connected cycle-subpaths from Rℓ1,ℓ2 (m, 1) to Rℓ1,ℓ2+1 (m, 1) defined as the sequence of those configurations belonging to Bl ℓ1,ℓ2 (m, 1) for any l = 1, . . ., ℓ 2 − 1. Enlarging the 1cluster, at a certain point, the process arrives in a configuration in which this cluster is either a vertical or a horizontal strip, i.e., it intersects one of the two sets defined in (4.21)- (4.22).If the descent arrives in S v pos (m, 1), then it proceeds by enlarging the vertical strip column by column.Otherwise, if it arrives in S h pos (m, 1), then it enlarges the horizontal strip row by row.In both cases, starting from a configuration with an 1-strip, i.e., a local minimum in M 2 pos by Proposition 5.3, the path exits from its cycle by adding a unit protuberance with a spin 1 adjacent to one of the two vertical (resp.horizontal) edges and increasing the energy by 2 − h.Starting from the trivial cycle given by this configuration with an 1-strip with a unit protuberance, the standard cascade enters a new cycle and it proceeds downhill by filling the column (resp.row) with spins 1.More precisely, the standard cascade visits K − 1 (resp.L − 1) configurations such that each of them is defined by the previous one flipping from m to 1 a spin m with two nearest-neighbors with spin m and two nearest-neighbors with spin 1.Each of these spin-updates decreases the energy by h.The process arrives in this way to the bottom of the cycle, i.e., in a configuration in which the thickness of the 1-strip has been enlarged by a column (resp.row).Starting from this state with the new 1-strip, we repeat the same arguments above until the standard cascade arrives in the trivial cycle of a configuration σ with an 1-strip of thickness L − 2 (resp.K − 2) and with a unit protuberance.Starting from {σ}, the process enters the cycle whose bottom is 1 and it proceeds downhill either by flipping from m to 1 those spins m with two nearest-neighbors with spin m and two nearest-neighbors with spin 1, or by flipping to 1 all the spins m with three nearest-neighbors with spin 1 and one nearest-neighbor with spin m.The last step flips from m to 1 the last spin m with four nearest-neighbors with spin 1.Note that if the vtj-connected cycle path (C 1 , . .where 4ℓ * is the perimeter of the cluster of spins 1 in ωk * .We want to show that it is not possible to have a configuration with k * spins 1 in a cluster of perimeter smaller than 4ℓ * .Since the perimeter is an even integer, we suppose that there exists a configuration belonging to V 1 k * such that the 1-cluster has perimeter 4ℓ * − 2. Since 4ℓ * − 2 < 4 √ k * , where √ k * is the side-length of the square √ k * × √ k * of minimal perimeter among those in R 2 of area k * , using that the square is the figure that minimizes the perimeter for a given area, we conclude that there is no configuration with k * spins 1 in a cluster with perimeter strictly smaller than 4ℓ * .Hence, ωk * ∈ F (V 1 k * ) and (5.11) is satisfied thanks to (A.8).
A.2 Additional material for Subsection 6.  pos .Hence, there exists i ∈ {0, . . .n} such that ω i ∈ D m pos .Thanks to (6.5) and to (6. 19) we have that the energy value of any configuration belonging to F (D m pos ) is equal to the min-max reached by any optimal path from m to X m pos \{m}.Thus, we conclude that ω i ∈ F (D m pos ).

Figure 1 :
Figure 1: Examples of configurations which belong to R3,6 (r, s) (a), B42,6 (r, s) (b).We color white the vertices whose spin is r and we color gray the vertices whose spin is s.

Theorem 4 . 6 (
Minimal gates for the transition from m to X m pos \{m}).Let m ∈ X m pos .If the external magnetic field is positive, then the following sets are minimal gates for the transition from m to X m pos \{m} (a) W pos (m, X s pos ),

Corollary 4 . 3 .
If the external magnetic field is positive, then for any m ∈ X m pos , (a) lim β→∞ P β (τ m Wpos(m,X s pos 1) : σ has a vertical 1-strip of thickness at least ℓ * with possibly a bar of length l = 1, . . ., K on one of the two vertical edges}, (4.21) S h pos (m, 1) := {σ ∈ X (m, 1) : σ has a horizontal 1-strip of thickness at least ℓ * with possibly a bar of length l = 1, . . ., L on one of the two horizontal edges}.(4.22)

c 2 c 10 (
a) Two vertical bridges on columns c 2 and c 10 .

r 2 (
b) An horizontal bridge on row r 2 .

r 6 c 4 (
c) A cross on column c 4 and on row r 6 .

Figure 4 :
Figure 4: Example of configurations on a 8 × 11 grid graph displaying a vertical s-bridge (a), a horizontal s-bridge (b) and a s-cross (c).We color black the spins s.

Figure 5 :
Figure 5: Stable tiles centered in any v ∈ V for a q-Potts configuration on Λ for any m, r, s, t ∈ S\{1} different from each other.The tiles are depicted up to a rotation of α π 2 , α ∈ Z.

M 1 Proposition 5 . 3 (
pos := {1, 2, . . ., q}; M 2 pos := {σ ∈ X : σ has strips of any spin m ∈ S of thickness larger than or equal to two}; M 3 pos := {σ ∈ X : σ has 1-rectangles that are not interacting with sides of length larger than or equal to two, either in a sea of spins m or in an m-strip for some m ∈ S\{1}};M 4 pos := {σ ∈ X : for any r ∈ S, σ is covered by r-rectangles.Every r-rectangle with r = 1 has minimum side of length larger than or equal to two, while every 1-rectangle has minimum side of length larger than or equal to one} ∪ {σ ∈ X : for any m, r, s ∈ S different from each other, σ has an r-strip of thickness one and adjacent to an m-strip and to an s-strip}; M 1 pos := {σ ∈ X : for any m, r ∈ S\{1}, σ has an m-rectangle with two consecutive sides adjacent to two r-rectangles and the sides on the interfaces are of different length}.Sets of local minima and of stable plateaux).If the external magnetic field is positive, then

Figure 8 :
Figure 8: Example of a path ω := (ω 0 , . .., ω n) such that H pos (ω i ) = H pos (ω j ), for any i, j = 0, . . ., n.Since all the configurations depicted have the same energy value and they are connected by means a path, they belong to a stable plateau.

Figure 11 :
Figure 11: Pictorial illustration of Step 6.The white rectangle denote a 1-rectangle of minimum side of length 1.

Figure 12 :
Figure 12: Example of configurations belonging to M pos .We color white the vertices with spin 1 and with the other colors the vertices with spins different from 1.

Figure 13 :
Figure 13: Illustration of three examples of σ ∈ D m pos when ℓ * = 5.We color black the vertices with spin m.In (a) the ℓ * (ℓ * − 1) + 1 = 21 spins different from m have not all the same spin value and they belong to more clusters.In (b) these spins different from m have the same spin value and they belong to three different clusters.In (c) the spins different from m have the same spin value and they belong to a single cluster.

Lemma 6 . 1 .
If the external magnetic field is positive, then for any m ∈ {2, . . ., q} the minimum of the energy in D m pos is achieved by those configurations in which the ℓ * (ℓ * − 1) + 1 spins different from m are 1 and they belong to a unique cluster of perimeter 4ℓ * .More precisely, F (D m pos ) = {σ ∈ D m pos : σ has all spins m except those in a unique cluster C 1 (σ) of spins 1 of perimeter 4ℓ * }. (6.4)Moreover, H pos (F (D m pos )) = H pos (m) + Γ pos (m, X s pos ) = Φ pos (m, X s pos ).(6.5)

Corollary 6 . 1 .
Let m ∈ X m pos and let ω ∈ Ω opt m,X s pos .If the external magnetic field is positive, then ω ∩ F (D m pos ) = ∅.Hence, F (D m pos ) is a gate for the transition m → X s pos .

Figure 14 :
Figure 14: Examples of σ ∈ Dm pos (a) and of σ ∈ Dm pos (b) and (c) when ℓ * = 5.We associate the color black to the spin m, the color white to the spin 1.The dotted rectangle represents R(C 1 (σ)).Figure (d) is an example of configuration that does not belong to Dm pos .
Figure 14: Examples of σ ∈ Dm pos (a) and of σ ∈ Dm pos (b) and (c) when ℓ * = 5.We associate the color black to the spin m, the color white to the spin 1.The dotted rectangle represents R(C 1 (σ)).Figure (d) is an example of configuration that does not belong to Dm pos .

Proposition 6 . 4 .
Let A pos be the habitat defined in (6.25).Then, F (A pos ) = X s pos and V (A pos ) = Γ m pos .Moreover, for any ω ∈ Ω opt m,X s pos , during the transition from any m ∈ X m pos to X s pos the process does not exit almost surely from A pos , i.e., lim β→∞ P β (τ m X s pos < τ m ∂Apos ) = 1.(6.26)

z∈X m pos W
pos (z, X s pos ) ⊆ G pos (m, X m pos \{m}), and we conclude exploiting [49, Theorem 5.1] and showing that any η ∈ S pos (m, X m pos \{m})\ z∈X m pos W pos (z, X s pos ) (6.32)

A. 2 . 1 2
Proof of Corollary 6.Every path from m ∈ X m pos to the other metastable configurations in X m pos \{m} has to pass through the set V m k := {σ ∈ X : N m (σ) = k} for any k = |V |, . . ., 0. In particular, given k * := ℓ * (ℓ * − 1) + 1, any ω = (ω 0 , . . ., ω n ) ∈ Ω opt m,X m pos \{m} visits at least once the set V m |Λ|−k * ≡ D m 4.1 (Maximum depth of a cycle in X \X s pos ).Consider the q-state Potts model on a K × L grid Λ, with periodic boundary conditions and with positive external magnetic field.
15))Hence, for any m ∈ S, if v has spin m and it has four nearest neighbors with spins different from m, then the tile centered in v is not stable for σ since the energy difference (2.8) is always strictly negative.Case 2. Assume that v ∈ V has three nearest neighbors with spin value different from m in σ, i.e., n m (v) = 1.Then, in view of the energy difference (2.8), for any m ∈ S and r / ∈ {1, m}, by flipping the spin on vertex v from m to r we have [40,d Theorem 4.4.Proof of Corollary 4.1.By [53,Lemma 3.6]we have that Γ pos (X \X s pos ) is the maximum energy that the process started in η ∈ X \X s Hence, let us proceed to estimate Γ pos (η, X s pos ) for any η ∈ X \X s pos .Let m ∈ X m pos .Note that for any η ∈ X \(X s pos ∪ X m pos ) there are not initial cycles C η While for any z ∈ X m pos \{m}, the initial cycles C z By this fact, that holds since we are in the metastability scenario as in [53, Subsection 3.5, Example 1], we get that for any m ∈ X m Proof of Theorem 4.4.The theorem follows by[40, Theorem 2.3].In order to apply this result it is enough to show that the pair (m, G) verifies the assumption pos has to overcome in order to arrive in X s pos = {1}, i.e., Γ pos (X \X s pos ) = max η∈X \X s pos Γ pos (η, X s pos ).(5.42) pos ).More precisely, we have that there exists k 1 > 0 such that for any β sufficiently large .37)Since ℓ ≤ ℓ * , comparing (6.36) with (6.37), we get that η ℓ−1 ∈ F (∂C(η 1 )), and (6.35) is verified.Let us now consider for any m ∈ S\{1} the local minimum ζ 1 ∈ Rℓ,ℓ (m, 1) ⊂ M 3 pos with ℓ ≤ ℓ * − 1. Arguing similarly to the previous case, we verify (6.34) by proving that Proof.For any m ∈ S\{1}, let η 1 ∈ Rℓ1,ℓ2 (m, 1) with ℓ * ≤ ℓ 1 ≤ ℓ 2 .By Proposition 5.3, η 1 ∈ M 3 pos is a local minimum for the Hamiltonian H pos .Using (4.13), our aim is to prove the following Lemma 6.4.For any m ∈ S\{1}, consider the local minimum η ∈ Rℓ1,ℓ2 (m, 1) with min{ℓ 1 , ℓ 2 } ≥ ℓ * .Let C(η) be the non-trivial cycle whose bottom is η.Thus,