Zero-temperature stochastic Ising model on planar quasi-transitive graphs

We study the zero-temperature stochastic Ising model on some connected planar quasi-transitive graphs, which are invariant under rotation and translation. The initial spin configuration is distributed according to a Bernoulli product measure with parameter $ p\in(0,1) $. In particular, we prove that if $ p=1/2 $ and the graph underlying the model satisfies the planar shrink property (which causes each finite cluster to shrink to a site and then vanish with positive probability) then all vertices flip infinitely often almost surely.


Introduction
In this paper, we deal with the zero-temperature stochastic Ising model (σ t ) t≥0 on some connected planar quasi-transitive graphs with homogeneous ferromagnetic interactions (see e.g.[17,27]), i.e. all the interactions are equal to a positive constant.The initial spin configuration is distributed according to a Bernoulli product measure with parameter p ∈ (0, 1), see e.g.[16,26,27].The dynamic evolves in the following way: each vertex, at rate 1, changes its spin value if it disagrees with the majority of its neighbours and determines its spin value by a fair coin toss in case of a tie between the spins of its neighbours.This process is often referred to as domain coarsening or majority dynamics and it is sometimes used as an opinion model.
A question of particular relevance is whether for each vertex v its spin flips only finitely many times almost surely, i.e. in other words whether σ t has an almost sure limit.We say that a vertex v fixates if the spin at v flips only finitely many times.According to the classification given in [17], a model is of type I if no site fixates almost surely, i.e all sites flip infinitely often a.s.; a model is of type F if all sites fixate almost surely, i.e. all sites flip only finitely many times a.s. and it is said of type M if there are both vertices that fixate and vertices that do not fixate almost surely.
The literature in the early years focused on the cubic lattice Z d and mainly with d = 2.It is known that the zero-temperature stochastic Ising model on Z with homogeneous ferromagnetic interactions is of type I for any initial density p ∈ (0, 1) (see [1,27]).
The disordered model on Z d , if the interactions {J x,y } are independent random variables with continuous distribution, is of type F (see [13,27]).Moreover, in d = 2 the homogeneous ferromagnetic model is of type I (see [27]).In [17], an analysis of the zero-temperature stochastic Ising model on Z d with nearest-neighbour interactions distributed according to a measure µ J (disordered model) is performed.In particular, it is proved that if the interactions are i.i.d.taking only the values ±J then the two dimensional model is of type M.An analogous result for d > 2 with a temperature fast decreasing to zero is obtained in [7].On the cubic lattice Z d , if initial configuration is distributed according to a Bernoulli product measure with parameter p sufficiently close to 1 (i.e. if p > p ⋆ d ), then the model is of type F, in particular each vertex fixates at the value +1 (see [16]).Moreover in [26] it is shown that p ⋆ d → 1/2 as d → ∞.For homogeneous trees of degree at least 3 and p sufficiently close to 1, it has been shown that the model is of type F (see [6,15]).
In [9,11], the case in which one or infinitely many vertices are frozen is studied.The main result of the first paper is that for d = 2 the model, with infinitely many frozen vertices, is of type F. On the contrary, in the second paper the authors show that the model in d = 2 is of type I when only one spin is frozen.
For articles on the stochastic Ising model on graphs other than Z d see for example [5,7,8,10,12,18,19]; in particular in [5] it is shown that the zero-temperature Ising model on the hexagonal lattice is of type F and in [7] it is proved that it is not of type F if simultaneous spin flips are allowed.In [18] the authors studied the Dilute Curie-Weiss Model, i.e. the Ising Model on a dense Erdős-Rényi random graph, and proved that depending on the distribution of interactions there are different behaviors.
In this paper, we deal with connected planar quasi-transitive graphs.The quasitransivity of the graph will be given by the invariance under translations and rotations.We will show that, under mild assumptions, the only rotations to consider are those of an angle θ ∈ π 3 , π 2 , 2 3 π, π (see Lemma 3 and Theorem 1).Such a class of graphs includes, for instance, the square, the triangular and the hexagonal lattice.
Our first result on the zero-temperature stochastic Ising model (Theorem 2) shows that a necessary condition for the model to be of type I is that the underlying graph has the shrink property.For example, the hexagonal lattice does not have the shrink property and the model on this lattice is of type F (see [27]).Thus, we will focus on a class of graphs having the shrink property.Actually, for technical reasons, we will use a potentially stronger definition of the shrink property that is the planar shrink property.
Our main result (Theorem 4) shows that if p = 1/2 and the graph is invariant under rotations, translations and has the planar shrink property, then the model is of type I.
Here we briefly present the general strategy to prove this achievement.First we show two preliminary results on general attractive spin systems with initial density p ∈ (0, 1] (see .More precisely we show that, for an attractive system, if a spin fixates to +1 with positive probability then the probability that it is constantly equal to +1 for all times t ∈ [0, ∞) is positive.After the general analysis developed in Theorems 1-2 we specifically study the zero-temperature stochastic Ising model.First we show that, under the shrink property and the translationergodicity, the cardinality of any cluster grows to infinity almost surely (Theorem 3).By this preliminary result, we are able to show that the cluster at the origin will intersect the boundary of any finite region infinitely often for t ∈ [0, ∞) with probability one.As already mentioned, we consider a planar graph that is invariant under translations and rotations of θ ∈ π 3 , π 2 , 2 3 π .Then, we construct a planar regular region centered at the origin that has the same rotation invariance of the graph.By the FKG inequality and the rotation invariance of the region, the cluster in the origin will intersect all the sides of the regular region with a positive probability larger or equal to p cross .We stress that, for t growing to infinity, the quantity p cross does not depend on the size of the region.By these properties and by the previous results we show that any ball centered in the origin has its spins equal to +1 infinitely often with a probability larger of p cross (see .Thus, with probability at least p cross no site will be able to fixate at the value −1.Finally, by considering the initial density p = 1/2 and by Lemmas 2-9, we show that all sites flip infinitely often almost surely (see Theorem 4).
The plan of the paper is the following.In Section 2, we define the zero-temperature stochastic Ising model, introduce the underlying graph and present some general results on attractive systems that will be useful for our discussion (see Lemmas 1-3 and Theorem 1).In Section 3, the main result, Theorem 4, is stated and proved through some lemmas and theorems.In Section 4, we present an infinite class of graphs having the planar shrink property (see Theorem 5).We also provide examples of graphs that have and do not have the shrink property, cases where the Ising model is of type I in the first case, and of type either M or F, in the latter.

Preliminaries
In this section, we introduce the graph underlying the zero-temperature stochastic Ising model and present some preliminary results that will be useful for our discussion in the next section.In particular, in Subsection 2.1 we define the Markov process by the infinitesimal generator and by the Harris' graphical representation.Moreover, we state two general lemmas (Lemma 1 and Lemma 2) for Glauber attractive dynamics.In Subsection 2.2, we introduce some notation on graphs and define the collection of graphs in which we are interested.More precisely we present in Lemma 3 and Theorem 1 some properties of sets in R 2 that are translation and rotation invariant.In Subsection 2.3, we describe in detail the zero-temperature stochastic Ising model I(G, p), where G is the underlying graph and p is the initial density.Moreover, we prove Theorem 2, which shows that the shrink property is a necessary condition to obtain that the I(G, p)-model is of type I.

2.1.
Attractive spin systems.We now introduce the spin systems referring mainly to [24,Chapter 3].We consider a spin system (σ t ) t≥0 , which describes ±1 spin flips dynamics on a countable set of vertices V .The state space is Σ = {+1, −1} V .The value of the spin at vertex v ∈ V at time t will be denoted by σ t (v).We introduce the usual order relation ≤ on Σ: given two configurations σ, σ ′ ∈ Σ, we say that σ ≤ σ ′ if for each v ∈ V , σ(v) ≤ σ ′ (v).The spin system (σ t ) t≥0 evolves as a Markov process on the state space Σ with infinitesimal generator L t , which acts on local functions f , and defined as where t ≥ 0, c t (v, σ) is the flip rate of the spin at vertex v, and σ v ∈ Σ is defined in the following way: We assume that c t (v, σ) is a uniformly bounded non-negative function, which is continuous on σ and satisfies the condition The condition in (2) guarantees the existence of the Markov process with infinitesimal generator L t (see [24]).We take the process (σ t ) t≥0 right continuous.We say that a spin system is attractive if c t (v, σ) is increasing in σ when σ(v) = −1 and decreasing in σ when σ(v) = +1.We are in particular interested to study Glauber dynamics, for which the relation We write the flip rates in the form where A v is a subset of V .Under (3) and the assumption the process defined in (1) can be constructed by the Harris' graphical representation (see e.g.[20,22,23,24]), which we now describe.We consider a collection (P v ) v∈V of independent Poisson processes with rate 1 interpreted as counting processes.For each v ∈ V , let T v = (τ v,n : n ∈ N) be the ordered sequence of arrivals of the Poisson process P v , associated with the vertex v.The probability that there is a flip at vertex v at time t (conditioning on the event {t ∈ T v }) is equal to c t (v, σ t − ), where σ t − := lim s→t − σ s .For convenience, to describe these events in more detail, we can use a family of i.i.d.random variables (U v,n : v ∈ V, n ∈ N) distributed according to a uniform random variable in [0, 1] and such that if U v,n < c τv,n (v, σ τ − v,n ), then the spin at v flips at time τ v,n (see [22] and [24]).
Lemma 1 below is well known, but we present a proof in order to construct the coupling that will be used in the proof of Lemma 2.
Proof.By hypothesis the order relation is satisfied at the initial time.Hence, it is sufficient to consider a single arrival of the Poisson process, i.e.only a possible spin flip, in order to show that the order relation is maintained, i.e. we will show that = −1 does not occur.Let us explicitly construct the desired coulping.We use the same Poisson processes for the two systems, but different families of i.i.d.uniform random variables (U v,n : v ∈ V, n ∈ N) and (U ′ v,n : v ∈ V, n ∈ N) for (σ t ) t≥0 and (σ ′ t ) t≥0 respectively.For each v, v ′ ∈ V and n ∈ N, let us consider the stopping time τ v,n .Moreover, for v ′ ∈ V \ {v} let us define and of the independent sequences n independence follows by: where 0 < a < b < 1 and u 1 , . . .u n−1 ∈ (0, 1).This implies that the distribution of U ′ v,n and the conditional distribution of . The independence of different sequences of uniform random variables can be proved in a similar way.
Whenever there is an arrival of a Poisson process, for example at time t for the vertex v (i.e.t ∈ T v ), the following situations can arise: ) then both systems change the ) then in the system (σ t ) t≥0 the spin at v does not change its value, while in (σ ′ t ) t≥0 the spin flip occurs at v; if ) ≤ U v,n then both systems do not have the spin flip at v. In all these three situations the order relation is maintained.
) then in the system (σ t ) t≥0 the spin at v change its value, while in (σ ′ t ) t≥0 the spin flip does not occur at n then both systems do not have the spin flip at v. In all these three situations the order relation is maintained.
, by using the relation (3) for Glauber dynamics and by attractivity, one has that ).
Thus, in the system (σ t ) t≥0 the spin at v changes its value, while in (σ ′ t ) t≥0 the spin flip does not occur at v, maintaining the order relation.
), the spin at v in (σ t ) t≥0 does not change and therefore the order relation is maintained.
By previous cases and since σ 0 ≤ σ ′ 0 , one deduces that the order relation is maintained at any time.Hence Now, we give the following definition which will be used in the next Lemma 2.
Definition 1.We say that a vertex v fixates if the spin at v flips only finitely many times and we say that a vertex fixates from time zero if its spin never flips.
In the following Lemma 2, for Glauber attractive systems, we compare the probability that a spin fixates or fixates from time zero.Lemma 2. Consider a Glauber attractive dynamics (σ t ) t≥0 where σ 0 has density p ∈ (0, 1].If a vertex w fixates at the value +1 with positive probability, then the vertex w fixates at the value +1 from time zero with positive probability. Proof.We define T w := inf{s ≥ 0 : σ t (w) = +1 ∀t ≥ s}, where inf ∅ = +∞.We assume that P(T w < ∞) = ρ > 0 and we choose t such that P(T w < t) ≥ ρ/2.
We consider a spin system (σ t ) t≥0 , described through the Harris' graphical representation with the independent Poisson processes of rate 1 (P v : v ∈ V ) and the i.i.d.uniform random variables (U v,n : v ∈ V, n ∈ N), with initial configuration σ 0 distributed according to a Bernoulli product measure with parameter p. Now, we construct another system (σ ′ t ) t≥0 with the same distribution.We make a resampling (indipendently by all other random variables already introduced) of the spin at vertex w in the initial configuration, such that +1 with probability p −1 with probability 1 − p, and σ ′ 0 (u) = σ 0 (u), for each u ̸ = w.We define a new Poisson process P ′ w that after time t has the same arrivals of P w .In the interval [0, t], P ′ w is a Poisson process of rate 1 indipendent by P w .This new process, by independent increments property, is still a Poisson process of rate 1.With positive probability one has P ′ w ( t) = 0. Thus, by independence, with probability at least ρ 2 pe − t, the following three events occur: Whenever these three independent events occur, we define (U ′ v,n : v ∈ V, n ∈ N) as follows: • . If one of the three events in (7) does not occur, the uniform random variables for each v ∈ V and n ∈ N. Now, we suppose that the three events in (7) occur.By construction, we have that σ 0 ≤ σ ′ 0 .We show that for t ∈ [0, t], one has σ t ≤ σ ′ t .Since P ′ w ( t) = 0, then in the process (σ ′ t ) t≥0 the spin at w remains equal to +1 until time t; hence σ t (w) ≤ σ ′ t (w) for all t ≤ t.When there is an arrival of a Poisson process P v with v ̸ = w, by using the same coupling of Lemma 1, it follows that the desired order relation is maintained until time t.In particular, at time t, σt ≤ σ ′ t.Now, applying the result of Lemma 1 by considering t as initial time, it follows that σ t ≤ σ ′ t for each t ≥ 0. Hence, the vertex w fixates from time zero with probability at least ρ 2 pe − t > 0, concluding the proof.□ 2.2.Notations and basic properties of graphs.We begin this subsection by presenting Lemma 3 that applies to subsets of R 2 which are invariant by translations and rotations.with the purpose of applying it to connected planar infinite quasi-transitive graphs.Later in the subsection, we introduce some definitions and notation on the graphs (see e.g.[14] and [19]) and we present the graphs on which the zero-temperature stochastic Ising model will be constructed.
Let us denote by ∥ • ∥ the Euclidean norm and by B(x, r) the ball of radius r > 0 centered in x.For any S ⊂ R 2 and x ∈ R 2 , we define the translation of a set as Given θ ∈ (0, π], we say that S is invariant under rotation of θ if there exists a point O ∈ R 2 , which we assume to be the origin, such that R θ (S) ⊂ S, where R θ is the rotation in the plane with center O and angle θ.Lemma 3. Let x ∈ R 2 be a non-zero vector and R θ a rotation in the plane with center O and angle θ ∈ (0, π].If S ⊂ R 2 is a non-empty set such that • S has a finite number of points in any ball; Proof.By hypothesis S is non-empty.Thus, by S + x ⊂ S, there exists a point v ∈ S, with v ̸ = O.Regarding the rotation, we write θ = 2πα where α that contradicts the property that each ball in R 2 contains a finite number of points of S. Hence, α necessarily belongs to Q.
Let α ∈ Q, we write α = m n with m, n ∈ N coprime.By Bézout's lemma, there exist a, b ∈ Z such that am + bn = 1.Let us select an integer k ∈ N such that a + kn ∈ N. One has Hence, we can consider only the angles of the form θ = 2π 1  n , for n ∈ N. By applying (n − 1) times the rotation R 2π n one obtains R − 2π n (S) ⊂ S, therefore the rotations with rational α are surjective onto S and consequently also invertible on S. In particular, R 2πα (S) = S for any α ∈ Q.

Now we define
r := min{||w|| : w ∈ R 2 , S + w ⊂ S} that is well-posed because, by hypothesis, there exists x ∈ R 2 such that S + x ⊂ S and S has a finite number of points in any ball.Therefore there exists v0 ∈ R 2 such that S + v0 ⊂ S with ||v 0 || = r.Notice that, without loss of generality, one can assume that r = 1 and v0 = (1, 0).Let us observe that, by R 2π 1 n (S) = S, it follows S + vk ⊂ S, vk , one has that S − v0 ⊂ S. Therefore the translation with respect to v0 is surjective onto S and being also injective it is invertible on S. Therefore S ± vk = S for k = 0, . . ., n − 1.Hence, one has The norm of v1 − v0 is 2 − 2 cos( 2π n ).Since v0 is a minimal norm vector such that S + v0 = S, one has By ( 8) one obtains that cos 2π n ≤ 1 2 , which gives n ∈ {2, 3, 4, 5, 6}.In order to get the result, we need to show that n = 5 contradicts r = 1.In this regard, we consider S + v0 + v2 = S where v2 = (cos(4π/5), sin(4π/5)).Notice that ∥v 0 + v2 ∥ = 2 + 2 cos 4 5 π < 1, which contradicts r = 1.Therefore θ ∈ π 3 , π 2 , 2 3 π, π and this concludes the proof.□ Remark 1.We note that a set S satisfying the hypotheses of Lemma 3 must be countable.Notice that for x = (0, 1) and θ ∈ π 3 , π 2 , 2 3 π, π there exist examples of sets S such that S + x = S and R θ (S) = S.Moreover, there are examples where Let us now recall some definitions and notation on graph theory (see e.g.[14,19]).Let G = (V, E) be a graph, where V is the set of its vertices (or sites) and E ⊂ V × V is the set of its edges.The degree of a vertex v ∈ V , denoted by deg(v), is the number of neighbours of v, i.e. deg(v) and S ⊂ V , we denote by N S (v) the set of neighbours of v in S, i.e.N S (v) := u ∈ S : {u, v} ∈ E and we define the degree of v in S as deg S (v) := |N S (v)|.Given U ⊂ V , the induced subgraph G[U ] is the graph whose vertex set is U and whose edge set consists precisely of the edges {u, v} ∈ E with u, v ∈ U .A path connecting a vertex v to a vertex u is a non-empty graph P = (V (P ), E(P )), where E) is said to be connected if for any two vertices u, v ∈ V there exists a path connecting them.We say that U ⊂ V is connected if the induced subgraph G[U ] is connected.We denote by d G (u, v) the distance in G of two vertices u and v defined as the length of a shortest path connecting u to v. Given a subset U ⊂ V , we define the external boundary of U as the set ∂ ext U := {v ∈ V \ U : ∃u ∈ U s.t.{v, u} ∈ E}.Now, we provide the following definitions (see e.g.[19]).
Definition 4 (Quasi-transitive graph).A graph G = (V, E) is said to be quasitransitive if V can be partitioned into a finite number of vertex sets V 1 , . . ., V N such that for any i = 1, . . ., N and any u, v ∈ V i , there exists a graph automorphism mapping u on v. Now we introduce planar graphs, which play a central role in our paper.An arc is a subset of R 2 that is the union of finitely many segments and is homeomorphic to the closed interval [0, 1].The images of 0 and 1 under such a homeomorphism are the endpoints of this arc.If A is an arc with endpoints x and y, the interior of A is A \ {x, y} (see [14]).
A plane graph is a pair G = (V, E) that satisfies the following properties: (1) V ⊂ R 2 is at most countable; (2) every edge is an arc between two vertices; (3) different edges have different sets of endpoints; (4) the interior of an edge contains no vertex and no point of any other edge.
A graph G = (V, E) is said to be planar if it can be embedded in the plane, i.e. it is isomorphic to a plane graph G.The plane graph G is called a drawing of G or embedding of G in the plane R 2 .We can identify a planar graph with its embedding in R 2 .Similarly, we say that (2)-( 4) hold (see e.g.[12,14]).Given a plane graph G = (V, E) and a set S ⊂ V , let Conv G (S) := Conv(S)∩V , where Conv(S) denotes the convex hull of S. We now define the shrink and planar shrink property, which will be central to our future discussion.
Definition 5 (Shrink property).Given a graph G = (V, E), we say that G has the shrink property if for each subset S ⊂ V with finite cardinality, there exists u ∈ S such that deg V \S (u) ≥ deg S (u).
Given a line ℓ ∈ R 2 , we denote by H ℓ 1 ⊂ R 2 and H ℓ 2 ⊂ R 2 the closed half-planes having ℓ as boundary.Given a non-empty subset S ⊂ V and a line ℓ, we define Definition 6 (Planar shrink property).For a plane graph G = (V, E), we say that G has the planar shrink property when, for any non-empty set S ⊂ V and for any line ℓ, one has: i is non-empty and has finite cardinality, then there exists u ∈ S ℓ i such that deg V \S (u) ≥ deg S (u).We say that a planar graph G has the planar shrink property if there exists an embedding of G in the plane for which such a property holds.
For a plane graph, it is immediate to note that the planar shrink property implies the shrink property.
We are interested in a connected planar infinite graph G = (V, E) with a specified embedding in R 2 such that the following conditions hold: (C1): There exists a non-zero vector x such that V + x ⊂ V and for any Then we say that G is translation invariant with respect to the vector x.(C2): Then we say that G is rotation invariant with respect to R θ .(C3): Each ball in R 2 contains a finite number of vertices of G.
By Lemma 3, it follows that a graph satisfying conditions (C1), (C2) and (C3) has θ ∈ π 3 , π 2 , 2 3 π, π and the translations and rotations in (C1), (C2) are graph automorphisms.For reasons that will become clear in the following, we do not deal with θ = π.The translations and rotations provide a partition of V in classes, in any case the partition given in Definitions 3 and 4 can be finer than the one given by only translations and rotations.For θ = π, it is straightforward to exhibit an example where the number of classes is infinite.For instance, we can consider G = (V, E) where V = Z 2 and the edge set is It is immediate to note that G is invariant under translation with respect to the vector (1, 0) and is invariant under rotation of π in the origin but not of π/2.Moreover, the classes of G are C n = {(i, ±n) : i ∈ Z} for n ∈ N 0 (see Figure 9 in Section 4).We present the following result for plane graphs that are translation and rotation invariant.2.3.The I(G, p)-model.We consider the stochastic process (σ t ) t≥0 , which describes ±1 spin flips dynamics on an infinite graph G = (V, E) with ∆(G) < ∞.The state space is Σ = {+1, −1} V and the initial state is distributed according to a Bernoulli product measure with density p ∈ [0, 1] of spins +1 and 1 − p of spins −1.The process corresponds to the zero-temperature limit of Glauber dynamics for an Ising model with formal Hamiltonian where σ ∈ Σ.The definition ( 9) is not well posed for infinite graphs.For this reason, we introduce the changes in energy at vertex v ∈ V as The process (σ t ) t≥0 is a Markov process on Σ with infinitesimal generator having as flip rates It is immediate to notice that the stochastic process is well defined, indeed the supremum in (2) is bounded by ∆(G) (see [24]).We note that the process is a Glauber attractive dynamics.Furthermore, since ∆(G) < ∞, the flip rates in (10) satisfy the condition in (5) with A v = N V (v).Therefore, this process can be constructed by the Harris' graphical representation (see [20,22,23,24]).In the following, we will refer to this model as I(G, p)-model where G is the underlying graph and p is the density of the Bernoulli product measure for the initial configuration.Now, in Theorem 2, we show that the shrink property is a necessary condition to obtain that the I(G, p)-model is of type I.
Theorem 2. Let G = (V, E) be a graph with ∆(G) < ∞.If G does not have the shrink property then for any p ∈ [0, 1] the I(G, p)-model is not of type I.
Proof.First let us consider the case p ∈ (0, 1].Since G does not have the shrink property then there exists a finite subset S ⊂ V such that deg V \S (u) < deg S (u) for any u ∈ S.Moreover, one has Since deg V \S (u) < deg S (u) for every u ∈ S, no site in S can change the value of its spin if σ 0 (u) = +1 for each u ∈ S.This fact implies that P each vertex in S fixates at the value +1 from time zero ≥ p |S| > 0. Thus, for any p ∈ (0, 1] the I(G, p)-model is not of type I.
Let us now consider p = 0.In this case, all sites fixate from time 0 at the value −1 almost surely and the I(G, 0)-model is of type F. □ Remark 2. Note that Theorem 2 does not imply that the I(G, p)-model is of type F or M. In fact if the graph G is not invariant under translation it could happen that the model does not belong to any of the three classes F, M and I.
If G does not have the shrink property, it is possible to show examples in which the I(G, p)-model is respectively of type F or M. It is known (see [5,27]) that if G is the hexagonal lattice then the I(G, p)-model is of type F. Now, we show in Example 1 that if we consider G ∈ G(4) as in Figure 2 then for any p ∈ (0, 1) the I(G, p)-model is of type M. ( Hence, by ergodicity (see [25,27]), there exist vertices that fixate at the value +1 almost surely.Now, let z, w i ∈ V be the vertices such that deg(z) = 4 and deg(w i ) = 3 for i = 1, . . ., 8 as in Figure 2. Let E z be the event that the vertices u i and w i fixate respectively at the value +1 and −1 for each i = 1, . . ., 8. With the same argument used in (12), we deduce that P(E z ) ≥ p 8 (1 − p) 8 > 0.Moreover, conditioning on the event E z , whenever there is an arrival of the Poisson process P z the spin flip at z occurs with probability 1/2.Thus, by Lévy's extension of the Borel-Cantelli Lemmas, z flips infinitely often with positive probability.By ergodicity (see [25,27]), it follows that there exist vertices that flip infinitely often almost surely.Therefore, the I(G, p)-model is of type M.

Main results
In the following, given σ ∈ Σ and v ∈ V , we denote by C v (σ) the cluster at site v for σ, defined as the maximal connected subset of V such that v ∈ C v (σ) and for any u ∈ C v (σ) one has σ(u) = σ(v).Now, we present the following theorem, which is an extension of Proposition 3.1 in [4].
Theorem 3.For d ∈ N, take a I(G, p)-model, where p ∈ [0, 1] and G is a graph embedded in R d that is translation invariant with respect to d linearly independent vectors.Moreover, suppose that G has the shrink property and ∆(G) < ∞.Then, the size of the cluster at a vertex v ∈ V diverges almost surely as t → ∞, i.e.
Proof.We explicitly use the elements ω of the sample space Ω.We prove the theorem by contradiction.Hence, for a vertex v ∈ V , let us define the event By contradiction assumption we suppose P(A) > 0. By continuity of the measure there exists M > 0 such that P(A M ) > 0, where Then, for any ω ∈ A M , one can define (T k (ω)) k∈N such that and, for k ∈ N, one recursively defines Let F t be the σ-algebra generated by the process up to time t.It is immediate to note that T k is a stopping time with respect to the filtration (F t ) t≥0 for any k ∈ N. We consider F T k for any k ∈ N. We notice that for ω ∈ A M , since ∆(G) < ∞ and |C v (σ T k )| < M , the cluster C v (σ T k ) can be equal only to a finite number of sets of vertices.For each of these sets of vertices, by the shrink property there is an ordered finite sequence of clock rings and outcomes of tie-breaking coin tosses inside a fixed finite ball that would cause the cluster to shrink to a single site w ∈ V with d G (w, v) < M (w could, in principle, depend on C v (σ T k )).Then, since the vertex w would have all neighbours with opposite spins, it could have an energy-lowering spin flip (with change in energy equal to −2deg(w)) and the cluster would vanish with positive probability.We define From the previous statements in this proof, one has that there exists δ > 0 such that Now, by the Strong Markov property of the process, for any k ∈ N we have the following lower bound for almost every ω ∈ A M .Thus ∞ k=1 ξ k (ω) = ∞, for almost every ω ∈ A M .Now, by using the Lévy's extension of Borel-Cantelli Lemmas (see e.g.[28]) with the sequence of events (B M,k ) k∈N and the filtration (F T k ) k∈N , we have that ω ∈ Ω : where C has measure zero.Then Thus, there exists a vertex w with d G ( w, v) < M such that energy-lowering spin flips occur at w infinitely many times with positive probability.By translation invariance with respect to d linearly independent vectors, v 1 , . . ., v d , we obtain that the graph G is quasi-transitive.The classes of the graph are all represented inside the parallelogram spanned by the d vectors v 1 , . . ., v d .Now, the translationergodicity implies that there exists a positive density of vertices for which energylowering spin flips occur infinitely often almost surely.This fact contradicts Theorem 3 and related remark in [27] (see also Lemma 5 in [7]).This concludes the proof.□ Remark 3. Theorem 3 is a generalization of Proposition 3.2 in [4] for graphs G having the shrink property.In [4], the result was given only for the cubic lattice Z d that in particular has the shrink property.
In the following, we consider the I(G, p)-model having G ∈ G(a) for a ∈ {3, 4} and it is invariant under translation with respect to x.Without loss of generality we take x = (1, 0).Let us consider a vertex ṽ having minimal Euclidean distance from the origin O. Clearly, ṽ = O when O belongs to V .In the case ṽ ̸ = O we consider the two distinct vertices ṽ, R θ (ṽ); since G is a connected graph, we can select a connected finite set S ⊂ V such that ṽ, R θ (ṽ) ∈ S. Finally we define the set of vertices where θ(a) = 2π/a.By construction U is connected and R θ(a) (U ) = U .
. By translation invariance of G with respect to x = (1, 0), one has v 0 + x ∈ V 1 .Now we can select a connected set of vertices U 0 such that V 1 ∩ B(v 0 , 2) ⊂ U 0 .Finally we choose r 1 ≥ 2 such that U 0 ⊂ B(v 0 , r 1 ).We are ready to present the following lemma.
Lemma 4. For any G ∈ G(a) with a ∈ {3, 4} and for any L ∈ R + there exists a connected set of vertices We define the set of vertices . Now, we show that W L is connected.Since and by using the triangular inequality, one obtains The previous inequality and Let W L as in Lemma 4. One can select a cycle U L ⊂ W L ; we call f L,∞ its outer face and f L,0 its inner face.
Remark 4. We notice that, by translational invariance with respect to the vectors x = (1, 0) and ȳ = (cos θ(a), sin θ(a)), one has where we write a n ≍ b n to mean that there exist two positive constants c 1 and c 2 such that c 1 ≤ an bn ≤ c 2 for all n ∈ N.This implies that G ∈ G(a) is amenable for any a ∈ {3, 4, 6}.Under the assumptions of amenability of the graph, the translation invariance of the measure µ and finite-energy of the measure µ, it is known that the infinite cluster is at most one almost surely (see [2,3,19]).For the zero-temperature stochastic Ising model, it is not known whether the measure induced at time t has finite-energy property.Therefore, we are not able to prove the uniqueness of the infinite cluster at time t.Instead, if the temperature is positive and decreases to zero, one has the property of finite-energy (see [7]).In this last case one obtains the uniqueness of the infinite cluster.Now, given an integer q ∈ N and δ < 1 2q , we consider T 1+δ (a) \ T 1−δ (a) and we show that there exists a collection of balls (B(c i , 4/q) : i = 1, . . ., aq) such that: a. T 1+δ (a) \ T 1−δ (a) ⊂ aq i=1 B(c i , 4/q); b. for any i = 1, . . ., aq, the center c i belongs to ∂T 1 (a); c. for any i = 1, . . ., q and m = 0, . . ., a − 1 one has c i+mq = R mθ(a) (c i ).In particular, R θ(a) ( aq i=1 B(c i , 4/q)) = aq i=1 B(c i , 4/q).It is clear that such a construction exists, for example by taking the centers of the ball equally spaced.The chosen balls in this construction will be maintained also in the sequel.
Proof.For a fixed i = 1, . . ., q, let us consider (c i+kq ) k=0,...,a−1 .For a = 3, 4, we note that Conv({c i+kq : k = 0, . . ., a − 1}) is an equilateral triangle or a square.Therefore, since c i+kq ∈ ∂T 1 (a) one has Conv({c i+kq : k = 0, . . ., a−1}) ⊃ B(O, 1  2 ).Let us now consider the segment having u 0 ∈ B(c i , 4/q) and u 1 ∈ B(c i+q , 4/q) as its endpoints.The distance between this segment and the origin O can decrease at most of 4/q with respect to the distance between O and the segment having endpoins c i , c i+q (see Figure 3).Then one obtains that Conv({u 0 , . . ., u a−1 }) ⊃ B(O, 1  2 − 4 q ).□ As already announced, we do not deal with θ = π.Indeed if we consider a = 2 which corresponds to θ(a) = π, this statement is false because Conv({u 0 , u 1 }) would be a segment and there is no ball contained in it.From now on we take q ≥ 24 and hence 1  2 − 4 q ≥ 1 3 .
, where U is defined in (13); • there exists i ∈ {1, . . ., q} such that C(σ t ) ∩ B(Lc i+kq , 4L/q) ̸ = ∅ for each k = 0, . . ., a − 1.We denote by E t L with t ∈ R + 0 the event that an L-cross of +1 occurs at time t.Moreover, we define We define the set of vertices S L (t) := C(σ t ) ∩ W L , where the properties of W L are given in Lemma 4. The previous Lemma 5 shows that for each time t ∈ R + 0 in which an L-cross of +1 occurs (see Figure 4), one has In other words, Lemma 5 says that |Conv G (S L (t))| ≍ L 2 .Lemma 6.Consider the I(G, p)-model, where p ∈ [ 1 2 , 1) and G ∈ G.If G has the shrink property, then , where the event A L is defined in (14).
We now explain the strategy for proving Lemma 6.Let U ⊂ V as defined in (13).If the initial density p ≥ 1/2, then U is contained in a cluster of +1 with probability at least (1/2) |U | , at any time t.By Theorem 3, the size of this cluster diverges almost surely as t → ∞.Now, by FKG inequality and by rotation invariance, one obtains that lim inf t→∞ P(E t L ) > 0, i.e. the cluster satisfies the properties in Definition 7 with positive probability.Note that this lower bound does not depend on L. By Reverse Fatou Lemma, we obtain the same lower bound on P(A L ).We are now ready to prove the lemma.
Proof.Let U t be the event that all vertices in U have spin equal to +1 at time t.By Lemma 1, FKG inequality and Harris' inequality (see [24,21]), it follows that If U t occurs, then, since U is connected, at time t all vertices in U belong to a same cluster, we call it C U (σ t ).Moreover, let C(σ t ) be the cluster of G[V ∩T L+2r1 (a)] that contains U .We denote by C W L (t) the event that the cluster C(σ t ) intersects W L , i.e.
By ( 16) and ( 17), it follows that lim inf Now, we write W L = ∪ q i=1 ∪ a−1 k=0 P i L,k where P i L,k := W L ∩ B(Lc i+kq , 4L/q) for i = 1, . . ., q and k = 0, . . ., a−1.We define the event C L,i,k (t) := { C(σ t )∩P i L,k ̸ = ∅} for i = 1, . . ., q and k = 0, . . ., a − 1.Hence, we have Thus, by rotation invariance and by the union bound, we have where ī ∈ {1, . . ., q} is such that P C L, ī,0 (t) ∩ U t = max i=1,...,q P C L,i,0 (t) ∩ U t .We note that C L, ī,k (t) ∩ U t is an increasing event for k = 0, . . ., a − 1; therefore where the first inequality follows by FKG inequality and by rotation invariance, and the last inequality follows by (19).We also notice that, by definition of P i L,k , one has q) ̸ = ∅}.Thus, by Definition 7, we have and hence Now, by (18), (20)   Now we give a simple definition that will be useful when related to E t L through Lemma 5.For t ∈ R + 0 , let F t L be the event that all sites belonging to B(O, L 3 ) have spin equal to +1 at some time s ∈ (t, t + 1).
Proof.Let σ ∈ Σ and (σ s ) s≥0 be the I(G, p)-model such that {σ t = σ} ⊂ E t L .We define another zero-temperature stochastic Ising model (σ ′ s ) s≥t with infinitesimal generator having the flip rates as in (10) and such that where C(σ t ) is the cluster of G[V ∩T L+2r1 (a)] in the configuration σ, as in Definition 7. By definition of σ ′ t and E t L , C(σ t ) ⊃ U is the unique cluster of +1 sites in the configuration σ ′ t .In configuration σ ′ t , we have that , where D i (σ ′ t ) for i = 1, . . ., k are clusters of −1 sites (we stress that k < ∞ because V ∩ T L+2r1 (a) has finite cardinality).
We notice that, by planarity of G, for each i = 1, . . ., k we have that ∂ ext D i (σ ′ t ) ⊂ C(σ t ).By planar shrink property, for each i = 1, . . ., k there exists an ordered finite sequence of updates (i.e. of clock rings and outcomes of tie-breaking coin tosses inside V ∩ T L+2r1 (a)) that would cause all sites of D i (σ ′ t ) ∩ Conv G (S L (t)) (see definition above formula (15)) to have spin equal to +1 in some time s ∈ (t, t + 1) with positive probability.Therefore, we get an ordered finite sequence of updates inside V ∩ T L+2r1 (a) that would cause all sites of Conv G (S L (t)) to have spin equal to +1 in σ ′ s (with s ∈ (t, t+1)), but since σ ′ t ≤ σ t this sequence of updates works, by the coupling in Lemma 1, in the same way for the original process (σ s : s ∈ (t, t+1)).Moreover, by Lemma 5, one has V ∩ B(O, L 3 ) ⊂ Conv G (S L (t)).Thus, there exists ϵ L > 0 such that We recall that A L := lim sup t→∞ E t L .Now, we define B L := lim sup t→∞ F t L .We are ready to present the following lemma.Proof.In the proof we will explicitly use the elements ω of the sample space Ω.First, let ω ∈ A L , i.e. an L-cross of +1 occurs infinitely often.Then one can define (T k (ω)) k∈N such that T 1 (ω) = inf{t ≥ 0 : E t L occurs}, and, for k ∈ N, we recursively define T k+1 (ω) = inf{t ≥ T k + 1 : E t L occurs}.Let F t be the σ-algebra generated by the process up to time t.It is immediate to note that T k is a stopping time with respect to the filtration (F t ) t≥0 for any k ∈ N. We consider F T k for any k ∈ N. By the Strong Markov property of the process and by Lemma 7, for any k ∈ N we have the following lower bound for almost every ω ∈ A L .Now, by using the Lévy's extension of Borel-Cantelli Lemmas (see e.g.[28]) with the sequence of events (F T k L ) k∈N and the filtration (F T k ) k∈N , we have that where C has measure zero.Then, by Lemma 6, we get Hence for all ϵ > 0 there exists a time s > t 1 + 1 such that

□
Let F be the σ-algebra generated by the process (σ t ) t≥0 .All the events introduced belong to F. Given a non-zero vector v such that G is translation invariant with respect to v, we define the configuration translated with respect to v as Let X be a F-measurable random variable.Then X = f ((σ t ) t≥0 ) where f is a measurable function.We define If X is an indicator function then f takes only the values 0 or 1.Let A ∈ F, one can define In the following result we will apply the ergodic theorem.We note that these processes are ergodic with respect to the translation if the initial conditions are given for instance by a Bernoulli product measure, see e.g.[20,24,25] and references therein.Now, we introduce some notation, which we will use in the proof of Theorem 4. For v ∈ V and t ∈ R + , let ) the event that the vertex v fixates at the value +1 (resp.−1) from time zero.Clearly, A ± v (t) ⊂ A ± v (t ′ ), for any t ′ ≤ t.We recall that {V 1 , . . ., V N } is the partition of the vertex set V that comes from the quasitransitivity of G ∈ G, see Theorem 1.We note that, since G is quasi-transitive, P(A ± v (t)) depends only on the class to which the vertex v belongs and does not depend explicitly on the vertex itself.Thus, for p = 1/2, for each i = 1, . . ., N , v ∈ V i , and t ∈ R + ∪ {∞}, we set ).The last equality follows by symmetry under the global spin flip for p = 1/2.Now, we are ready to prove our main result.Proof.We will prove the theorem by contradiction.Suppose that there exists j ∈ {1, . . ., N } such that ρ j (∞) > 0 that by Lemma 2 is equivalent to have a site that fixates with positive probability.We choose the following constants: ϵ = 4 p cross and ε = 1 8 .We notice that, by continuity of the measure, the limit of ρ i (t) as t → ∞ exists and is equal to for each v ∈ V i .This implies that there exists a time t ϵ > 0 such that Since G ∈ G, there exist two linearly independent vectors x1 and x2 such that G is translation invariant with respect to them.We want to construct on the graph G disjoint regions of a suitable size L 0 centered in n 1 x1 + n 2 x2 with n 1 , n 2 ∈ Z.By ergodicity (see [20,25,27]), one has lim r→∞ 1 n(r, j) v∈B(O,r)∩Vj where n(r, j) := |B(O, r) ∩ V j |.Thus, (24) implies that there exists r ∈ R + such that P 1 n(r, j) v∈B(O,r)∩Vj Then, in particular P v∈B(O,r)∩Vj Now, we construct disjoint regions of size L 0 on the graph G in the following way.

Theorem 1 .Figure 1 .
Figure 1.The graph on the left belongs to G(4), hence it is invariant under rotation of π/2.The graph on the right is only invariant under rotation of π.

Figure 2 .
Figure 2. Example of a graph G ∈ G(4) that does not have the shrink property and, for p ∈ (0, 1), the I(G, p)-model is of type M.

For a = 3 ,
4 one respectively obtains that T L (a) is an equilateral triangle or a square.Now, let us consider the class V 1 ⊆ V (see Theorem 1 and Definition 4).If the graph G

Figure 3 .
Figure 3.The distance between the segment having u 0 ∈ B(c i , 4/q) and u 1 ∈ B(c i+q , 4/q) as its endpoints and O can decrease at most of 4/q (the length of the radius) with respect to the distance between the segment having endpoins c i and c i+q and O.

Figure 4 .
Figure 4. Example of a realization of an L-cross.Now, we present the following definition.Definition 7 (L-Cross).Given G ∈ G(a) with a ∈ {3, 4}, we say that an L-cross of +1 occurs at time t if there exists a cluster C(σ t ) of G[V ∩ T L+2r1 (a)] such that

Lemma 7 .
Consider the I(G, p)-model, where p ∈ [ 1 2 , 1) and G ∈ G.If G has the planar shrink property, then there exists ϵ L > 0 such that

Theorem 4 .
If G = (V, E) ∈ G has the planar shrink property, then the I(G, 1/2)model is of type I, i.e., all sites flip infinitely often almost surely.

Figure 6 .
Figure 6.Examples of graphs in G(4) that do not have the shrink property.

Figure 7 .
Figure 7. On the left the triangular lattice, example of a graph G ∈ G(6) ∩ H. On the right a double triangular lattice, example of a graph G ∈ (G(3) \ G(6)) ∩ H.

Figure 9 .
Figure 9. Example of a graph G ∈ H that is invariant under translation and rotation of π, but not of π/2.The number of classes of G is infinite.
By Theorem 3, we have that lim t→∞ |C U (σ t )| = ∞ almost surely.Thus, by planarity of G and V ∩ f L,0 has finite cardinality (see Remark 4), we get lim Lemma 9.For any L ∈ R + and t 1 ≥ 0, one has P D(L; t 1 , ∞) ≥ p cross .