Stochastic Ising model with flipping sets of spins and fast decreasing temperature

This paper deals with the stochastic Ising model with a temperature shrinking to zero as time goes to infinity. A generalization of the Glauber dynamics is considered, on the basis of the existence of simultaneous flips of some spins. Such dynamics act on a wide class of graphs which are periodic and embedded in $\mathbb{R}^d$. The interactions between couples of spins are assumed to be quenched i.i.d. random variables following a Bernoulli distribution with support $\{-1,+1\}$. The specific problem here analyzed concerns the assessment of how often (finitely or infinitely many times, almost surely) a given spin flips. Adopting the classification proposed in \cite{GNS}, we present conditions in order to have models of type $\mathcal{F}$ (any spin flips finitely many times), $\mathcal{I}$ (any spin flips infinitely many times) and $\mathcal{M}$ (a mixed case). Several examples are provided in all dimensions and for different cases of graphs. The most part of the obtained results holds true for the case of zero-temperature and some of them for the cubic lattice $\mathbb{L}_d=(\mathbb{Z}^d, \mathbb{E}_d)$ as well.


Introduction
In this paper we deal with a class of non homogeneous Markov processes (σ(t) : t ≥ 0) in the frame of random environment. In particular, we consider a generalization of Glauber dynamics of the Ising model, see e.g. [18], in the case of flipping sets whose cardinality is smaller than or equal to a given k ∈ N. However, accordingly to Glauber, also our dynamics will satisfy the reversibility property with constitute the random environment. The case α = 1 (resp. α = 0) corresponds to the interactions of the homogeneous ferromagnetic (resp. antiferromagnetic) Ising model.
The temperature profile of the model is a function T : [0, ∞) → [0, ∞), which is measurable with respect to the Borel σ-algebra B(R) with T (t) denoting the temperature at time t, for any t ∈ [0, ∞). In most of the results it is assumed that lim t→∞ T (t) = 0 with a specific reference to T fast decreasing to zero, this will be important due to its connections with the well studied case of T ≡ 0.
In the case of zero temperature stochastic Ising models with homogeneous interactions, the following question is of particular relevance: (Q) Does a given spin on v ∈ V flip infinitely many times almost surely?
However, for constant and positive temperature T , question (Q) does not make sense because all the spins flip infinitely many times. Moreover, question (Q) should be rephrased also in the frame of random interactions. In fact, it is not always meaningful to deal with single sites, because the random environment leads to sites differently behaving.
In the deep contribution of Gandolfi, Newman and Stein [14] the authors propose also a classification for these models, which is a partition of them. Specifically, a model is of type I, F, and M, according to if all the sites flip infinitely many times (a.s.), finitely many times (a.s.), or some sites flip infinitely many times and the others do it finitely many times (a.s.), respectively. We adopt here the same classification.

(Q') Is a model of type I, F or M?
We notice that F corresponds to the almost surely convergence with respect to the product topology.
Vice versa I and M correspond to a.s. no-convergence. In this paper we show that some universal classes can be identified, in the sense that the graph G and the parameter k determine the class of the model under mild conditions on α, γ, T . Indeed, the main part of our results holds true for α ∈ (0, 1) and γ ∈ [0, 1] under some natural requirements on the decay rate of the temperature profile T .
In a very different context, [1] has shown the recurrence of annihilating random walks on Z d , with d ∈ N, under very general conditions. This paper has as a consequence that the one-dimensional stochastic Ising model with α = 1, γ ∈ (0, 1) and T ≡ 0 is of type I (see [23]).
In [14] there is an analysis of the zero-temperature case for the cubic lattice L d = (Z d , E d ), γ = 1/2 and different product measures µ J over R for the interactions J . Among the results, the authors provide a complete characterization for L 1 and for all the measures µ J , i.e. they identify the classes I, F, and M for each µ J . In doing so, [14] adapts and extends [1] in this context. Moreover, the authors identify the class M for L 2 when the measure µ J is a product of Bernoulli with parameter α ∈ (0, 1).
In [23] there is a treatment of α = 1 and γ = 1/2 for L 2 , where it is proven that the model is of type I. Under the same conditions, [2] refines [23] in discussing the recurrence and the growth of some geometrical structures.
It is also worth noting that [23] analyzes the framework where µ J is a continuous measure over R with finite mean, and they find F. The finite mean restriction has been eliminated in the same setting by [11], and the identified class remains F.
The contribution of [13] is for T ≡ 0, α = 1, L d with d ≥ 2 and γ > γ d , with γ d ∈ (0, 1). The authors show that, when γ > γ d , the value of any spin converges to +1 a.s., hence leading to a model of type F.
In this last context, in order to prove that, for a large value of γ, all the spins converge to +1, it is worth mentioning [3] where the stochastic Ising model at zero-temperature on a d-ary regular tree T d is analyzed. Analogously to [13,22], the authors prove that there existsγ d ∈ (0, 1) such that for γ >γ d all the spins converge to +1. Moreover it is also shown, along with other results, that lim d→∞γd = 1/2.
There are important results for the convergence of the system to the Gibbs state in the case of low temperature (see e.g. [6,7,9,20]) or high temperature (see e.g. [6,8,19,21]). In particular, in [6], in the framework of random interactions, it is shown a different approach to the equilibrium measure in relation to the temperature. We believe that there is a connection between the behaviour of the stochastic Ising model with α ∈ (0, 1) at low, constant and positive temperature and at zero-temperature. In this respect, it seems that the type of the model, F or M, at zero-temperature is related to properties of metastability of the same model at low constant temperature. The geometric structure of the graph is also relevant in both the cases of zero-and low constant temperature. In [6,14] some of the main results come out from a combinatorial-geometric interpretation of the underlying graphs.
For cases different from L d at zero-temperature, we mention [3] and [12], with tree-related graphs; [30], where trees and cylinders originating by graphs are considered; [16], where the hexagonal lattice is explored. In this latter paper the authors move from [23], where it is proven that sites fixate, and show that the expected value of the cardinality of the cluster containing the origin becomes infinite when time grows.
As already announced above, we aim to provide an answer to (Q') in the framework of general graphs.
In particular, we deal with graphs which are periodic and embedded in R d . In so doing, we contribute to the literature, which is mainly focussed on L 1 (see e.g. [14]), L 2 (see e.g. [2,14,23]) and the hexagonal lattice (see e.g. [16]) with the exceptions of the general dimensional cubic lattices L d in [13,22].
Notice that this topic is important either at a purely theoretical level as well as in the applied science.
Indeed, we mention [10,24,28,29], where applications of Ising models to social science are presented. In particular, [10] deals with Glauber dynamics at zero-temperature over random graphs, where nodes are social entities and the spatial structure captures the social connections among the nodes. For a review of the relevant contribution on the so-called sociophysics, refer to [28].
This paper adds to the literature on the stochastic Ising models as follows: (i) We allow for simultaneous flips of the spins, hence allowing for flipping regions. This statement has an interest under a theoretical perspective and it seems to be also reasonable for the development of real-world decision processes (think at the changing of opinions process of groups of connected agents rather than of single individuals). The cardinality of the flipping sets is constrained by a parameter k, which will be formalized below. Some material related to this aspect can be found in [30].
(ii) The considered graphs are periodic, infinite and with finite degree. This framework naturally includes L d , for each d ∈ N. This generalization allows us to provide theoretical results and physically consistent examples (like crystal lattices) outside the restrictive world of L d .
(iii) The temperature is taken not necessarily zero. Specifically, we consider a temperature fast decreasing to zero and we require only in one result its positivity. In doing so, we develop a theory on the reasonable situation of a temperature changing continuously in time, without assuming the jump from infinite to zero. The framework of temperature fast decreasing to zero includes also the case of T ≡ 0.
In more details, Lemma 1 is a technical result giving the framework we deal with. Specifically, it provides some grounding consequences of the definition of temperature profile fast decreasing to zero, which are useful in the rest of the paper. Theorem 1 gives sufficient conditions for the temperature profile having a similar or different behaviour of the zero-temperature case. It is shown that the model is of type I under an asymptotic condition for the temperature profile. This result depends on the Hamiltonian, and can then be rewritten for general Gibbs models endowed with a Glauber-type dynamics. Some conditions of Theorem 1 can be replaced with weaker ones over graphs with even-degree sites (e.g. L d ).
In Theorem 2 we present a result stating that the model is of type M or I, under some hypotheses.
Specifically, it is required that the temperature profile T is fast decreasing to zero, positive and the graph belongs to the rather wide class of k-stable d-Egraphs (see next section for the formal definition of this concept). Such a class contains also the cubic lattices L d , with d ≥ 2. We believe that Theorem 2 represents our main result because it is a relevant step to the identification of the type of stochastic Ising models in dimension d ≥ 2. We stress that the positivity of the temperature is required only in this result.
Theorem 3 states some conditions to obtain models of type M, and it is used to prove some findings in Section 4.
Theorem 4 shows that the model with ferromagnetic interactions and γ = 1/2 on the hexagonal lattice is not of type F when k ≥ 2. In so doing we complement [23], where it is proven that the same model with k = 1 and T ≡ 0 is of type F.
In Section 4 we describe conditions leading to fixating sites. Lemmas 4 and 5 highlight the connection between lowering and increasing energy flips. Indeed, under the hypothesis of Lemma 1, the number of such flips is a random variable having finite mean on any given set of flipping spins. These two Lemmas are widely used to prove the results of this Section. Theorem 5 is inspired by [14,23] and generalizes their outcomes to the case of k > 1. In particular, it links the parameter k with the properties of the graph to obtain that some sites fixate, i.e. the model is of type M or F. Theorem 6 formalizes the intuitive fact that if there exists a set of sites strongly interconnected and weakly connected with the complement of the set, then these sites will fixate with positive probability.
In Definition 4 we adapt to our setting the concept of e-absent configurations (namely, k-absent on J here), which has been introduced for the graph L 2 in [2,14]. We generalize such a definition by including k > 1 and the considered class of graphs. Theorems 7 and 8 are based on k-absence. The former result states that configurations which are k-absent on J can appear only on a random finite time interval (a.s.) when interactions are properly selected; the latter one provides a condition on the graph and the parameter k such that some sites have a positive probability to fixate. An interesting consequence of the definition of k-absence is that if a configuration is k-absent on J , then it is k -absent on J , for each k > k. Hence, a large value of k seems to facilitate the fixation of the sites (see Theorem 7). Differently, a large value of k is an obstacle for the fixation of the sites in Theorem 8. This contrast leads to a not straightforward link between the value of k and the identification of the model. However, in our setting, we have shown that a model on the hexagonal lattice is of type F for k = 1 (see also [23]) and it is not of type F for k ≥ 2 (see Theorem 4). This Theorem suggests a general conjecture that link the value of k with the type of the model (see the Conclusions).
The provided examples illustrate a wide part of the outcomes, and complement the theoretical findings of the paper. In particular, we have introduced the graph Γ ,m (G), it is constructed by replacing the original edges of a graph G = (V, E) with more complex structures (see Definition 5). For this specific class, we have provided some conditions for which the models over such graphs are of type M (see Theorem 9) and I (see Theorem 10). The case F is left as a conjecture in the Conclusions.
The last section of the paper concludes and offers some conjectures and open problems. In order to assist the reader, we have provided some figures presenting the main contributions of the related literature at zero-temperature, the results obtained in the present paper for the Γ ,m (G) graphs and for k-stable d-E graphs (see Figure 6).

Definitions and first properties of the model
The main target of this section is to define a dynamical stochastic Ising model. In order to do it we introduce some notation. i.e. the space R d is seen as the union of disjoint hypercubes [0, 1) d + z with z ∈ Z d . In order to specify the d-graph G = (V, E) it is enough to give the vertices V Cell inside Cell with the edges ; by periodicity and the fact that Cell contains finitely many vertices, then d G is finite.
If a d-graph has at least a vertex with even degree we denote it as d-E graph. As an example of a d-E graph we take the lattice L d = (Z d , E d ), where the degree of any vertex is equal to 2d.
Some standard definitions on graph theory are now recalled. Given n ∈ N, a path of length n starting in x, x 1 , . . . , x n−1 , x n = y) with x i ∈ A, for i = 0, . . . , n. We say that a graph G = (V, E) is connected if V is connected.
Since, without loss of generality, one can study separately the different connected components, as will be clear below, in the sequel we will consider only connected d-graphs.
The distance ν G (u, v) in G of two vertices u, v ∈ V is the length of the shortest path (not necessarily unique) starting in u and ending in v. For u ∈ V and L ∈ N we define the ball centered in u with radius L as whereas the external boundary is : v ∈ V ) to denote the configuration that corresponds to flip the configuration σ on the set A.
The formal Hamiltonian associated to the interactions J ∈ {−1, +1} E is However, the definition (1) is not well posed for infinite graphs. Thus, we shall work with the increment of the Hamiltonian at A, namely for a finite set A ⊂ V Notice that the value of ∆ A H J (σ) can be only an even integer. In particular, if ∆ A H J (σ) = 0, then 2.3. Dynamics. The dynamics of the system will be a non-homogeneous Markov process depending on the interactions, the temperature profile and the initial configuration. For k ∈ N, we call A k the collection of the connected subsets A ⊂ V having cardinality smaller or equal to k. It is important to stress that, for a given v ∈ V , the set {A ∈ A k : v ∈ A} is finite for a d-graph. Therefore, the set A k is countable and, following [18], one defines the infinitesimal generator related to k as follows In the case in which T (t) = 0 for t ≥ 0, set We remark that in the case of T ≡ 0 and k = 1 our dynamics is the Glauber dynamics at zero temperature (see for instance [2,11,14,22,23]).
The previous defined process can be constructed by using a collection of independent Poisson processes (P A : A ∈ A k ) with rate 1, the so-called Harris' graphical representation [15]. (σ(τ A,n )), then there is a flip at the set A at time τ A,n (see [15,18]).
The representation of the Markov process based on the Poisson processes is very popular in the framework of zero-temperature dynamics, since it exhibits remarkable advantages with respect to the one based on the generator. Firstly, the spatial ergodicity of the process with respect to the translations related to the canonical basis can be invoked on the ground of a very general theory (see [15,17,23,26]).
Secondarily, this representation is the natural setting for the proofs of the results.
We say that a flip at A at time t ∈ T A is in favour of the Hamiltonian if ∆ A H J (σ(t)) < 0; it is indifferent for the Hamiltonian if ∆ A H J (σ(t)) = 0 and it is in opposition of the Hamiltonian if For k ∈ N, t ∈ [0, ∞] and A ∈ A k , we define the sets S − A,t , S 0 A,t and S + A,t as and, for any x ∈ V , Moreover, we define the set of total flips involving the site x as and the set of total arrivals involving the site x as We now provide the definition of the probability measure associated to the dynamical model defined in (3), which can be written as where µ J is the probability distribution of the interactions J forming the quenched random environment; ν σ(0) is the probability distribution of the initial configuration σ(0); P k is the measure of the arrivals of the i.i.d. Poisson processes on the sets A ∈ A k and the sequences of independent U 's as in (5)independent also from the Poisson processes.
Some particular cases are important in our context. The considered measures over the interactions are of two families where Ber e (α) is the Bernoulli distribution with parameter α and support {−1, +1}, labeled by the edge e ∈ E. The deterministic case is Measure µ J in (10) is the case of ±J model in [14], while (11) is the case of deterministically selecting the interactions on the edges of the graph.
Analogously, for the initial configuration σ(0), we use To avoid a cumbersome notation, we will denote the probability measure P k,µ J ,ν σ(0) simply as P, and the identification of it will be clear from the context. Analogously, the expected value associated to the probability measure P k,µ J ,ν σ(0) will be indicated as E.
Let us consider a d-graph G = (V, E). Then, • when µ J and ν σ(0) are as in (10) and (12), respectively, then the process will be denoted as (k, α, γ; T )-model on G, for a temperature profile T ; • when µ J and ν σ(0) are as in (11) and (13), respectively, then the process will be denoted as The definition of the measure associated to the model allows us to introduce the quantities of interest in assessing its type.
By adopting the notation of [14], a model is said to be of type I, F, and M, if all the sites flip infinitely many times (a.s.), finitely many times (a.s.), or some sites flip infinitely many times and the others do it finitely many times (a.s.), respectively.
In particular, using standard ergodic arguments one can see that for a (k, α, γ; T )-model on a d-graph G the quantity does exist and it is constant almost surely, and it does not depend on the vertex v (for details see [14,23]).

Conditions for ρ I > 0
The first results are given in a general setting, where the interactions J and the initial configuration σ(0) are provided; moreover, they could be presented for a completely general Glauber dynamics associated to a Gibbs measure and, thus, associated to a Hamiltonian.
First, we give the following uniform integrability condition Notice that the condition of being fast decreasing to zero is an asymptotic property of the temperature profile, and the complete knowledge of the behavior of T is not required.
An immediate consequence of the previous definition is the following Furthermore, for any finite set V 0 ⊂ V , and P x∈V 0 Proof. Let us define Clearly, for any x ∈ V , K x is finite and also K is finite because it corresponds to take the maximum only inside Cell.
Since T is fast decreasing to zero, we can take t 0 sufficiently large to have One has and (14) is proved.
To prove (15) we notice that, by continuity of the measure and being T fast decreasing to zero, for and thus This leads to (15).
then the temperature profile T is fast decreasing to zero.
then N x is infinite almost surely.
Proof. We start by proving the first item of the theorem.

By (18) one has
We also consider a constant a ∈ (1, 1 Thus, there existsn ∈ N large enough such that, for any n ≥n, In fact, for any t ∈ [n − 1, n) using (4) and the fact that 1 1+e x decreases in x, one has By (20) and for a given r ∈ (r, 1), there exists t such that for any t > t then 4/T (t) > ln t/r . Now, it is sufficient to taken = t + 1.
The second inequality in (22) gives that, for any n ≥n and for any t By setting r = 1/a, we obtain (21). We set p n = 1 (n−1) a , for n ≥n.
Then, for t ≥n, where Bernoulli random variables with parameter p n independent from Q x,n . Notice that we are implicitly considering a common probability space for all the random variables. The last inequality in (23) comes from the graphical representation.
Using the definition of K given in (17), the proof of the first item of this theorem ends by noticing that the last term in (23) is smaller or equal than the last term tends to zero for t → ∞.
We give now the proof of the second item of the theorem.
Condition (19) gives For x ∈ V and n ∈ N, we define the event i.e. there is at least an arrive for the Poisson P {x} in the interval [n − 1, n). We consider the collection of independent and equiprobable events (F x,n : x ∈ V, n ∈ N). In particular, There existsn ∈ N large enough such that, for any n ≥n, it results In fact, by (4), one has Therefore, by (24), there exists t such that for t > t one has 4d x /T (t) < ln t. Analogously to the previous item, definen = t + 1.
Hence, for any n ≥n and for any t Then by Lévy's conditional form of the Borel-Cantelli lemma (see [31]), one obtains the proof of the statement.
In more details, let us consider n ∈ N and define G n as the σ-algebra generated by all the Poisson processes until time n and the related U 's. Hence, the process σ(·) is seen as a function of the Harris' graphical representation. Let us define the event Therefore, E n ∈ G n and P(E n |G n−1 ) ≥ 1−e −1 1+n and this implies that Formula (25) In fact, for each J ∈ {−1, +1} E and σ ∈ {−1, +1} V , the flips in opposition of the Hamiltonian have This statement can be generalized to the case of a site x with even degree d x . In , then one among hypotheses (26) and (19) is verified.
As an immediate consequence of Theorem 1 we obtain To the benefit of the reader, we now adapt Lemma 5 in [14] and related definitions to our context of nonhomogeneous Markov process.

Definition 2. Let us consider a continuous-time Markov process
Moreover, let us take a measurable set A ⊂ Z. The set A recurs with probability p ∈ [0, 1], if We also say that a measurable set

Lemma 2. Let us consider the process Z = (Z t : t ≥ 0) and the events A and B as in Definition 2. If
A recurs with probability p ∈ (0, 1] and then B recurs with probability p ≥ p. Proof. Let us define W = {{s > 0 : Z s ∈ A} unbounded}. If W occurs we can define recursively an infinite sequence of stopping times (T j : j ≥ 0) such that T 0 = 0 and with the convention that inf ∅ = ∞.
By the strong Markov property and formula (27), one obtains on W , for j ∈ N. By (28) and (29) The Lévy's extensions of the Borel-Cantelli Lemma implies that on W . Therefore B recurs at least with probability p.

Now we introduce a new condition on the d-E graphs.
Definition 3. Let us consider k ∈ N. We say that a d- Notice that if a d-E graph is k-stable, then it is k -stable, for any k < k, being A k ⊂ A k . We also notice that for k = 1 the second condition of Definition 3 is automatically satisfied.
An example of a k-stable d-E graph is the cubic lattice L d = (Z d , E d ), for each k ∈ N and d ≥ 2.
The case L 1 = (Z, E 1 ) is an example of a d-E graph that is not 1-stable, and thus it is not k-stable for each k ∈ N. In fact, the first condition in Definition 3 is not satisfied.
interactions J are taken as follows Proof. If A ∈ A k and v / ∈ ∂A ∪ ∂ ext A, then there is nothing to prove because all the interactions and spins involved in the computation of ∆ A H J (σ) are equal to +1.
Proof. By contradiction, we suppose that ρ I = 0.
In this proof we will use several stochastic Ising models sharing the same realized random configuration σ but with different interactions. Among them, we denote by original system the model on the k-stable In the graphical representation we denote the Poisson processes associated to the original model by (P A : A ∈ A k ), and the related uniform random variables by (U A,n : A ∈ A k , n ∈ N).
By the assumption that the d-E graph is k-stable, there exists a vertex v ∈ V with even degree satisfying the properties given in Definition 3. We select such a vertex. Let us consider the sets B 4k (v) and B 8k (v), where we are using the constant k given in the statement of the theorem. Define For M ∈ R + , let us introduce the event F (1) By hypothesis that ρ I = 0, Now, recall the concept of C) all the remaining interactions coincide with those of the realized interactions J of the original system.
The value Q is finite because we are dealing with d-graphs, and it depends on G, k and v.
All the quantities related to the q-th system are denoted, in a natural way, by adding, when needed, to the original quantities the superscript (q).
Recall that all the systems described above share the initial configuration σ of the original one.
Moreover, the collection of Poisson processes (P A : A ∈ A k ) are assumed to be common for all the Q systems until time M and, accordingly, the systems share also the same variables U 's defined in (5).
Given the q-th system, we define the set It is known that |D (q) | < ∞ a.s. (see e.g. [25]).
We introduce the random set For any M ∈ R + and M > M , we define Among all the different systems -described in A), B) and C)-we select the only one having interactions such that J e = σ x (M ) when e = {x, y} ∈ E with x ∈ ∂ ext B 4k (v) and y ∈ ∂B 4k (v). We denote such random interactions asJ . We call the corresponding system as capital system, this process is denoted by (Σ(t) : t ≥ 0). We assume that it is a stochastic Ising model, and we will check it below.
One has that F Observe that M .
Since (Σ(t) : t ≥ 0) will be proved to be a stochastic Ising model, we can use (15) We define (P A : A ∈ A k ) a set of Poisson processes with rate 1 that are active in the time interval Given A ∈ A k , we consider a new process P A whose set of arrivals T A is defined as follows: For each A ∈ A k , the set T A is distributed as a Poisson process of rate 1, and (P A : A ∈ A k ) is a set of independent Poisson processes with rate 1, which are also independent from the U 's and from theŪ 's.
The U 's (resp.Ū 's) are used when the arrivals of the Poisson processes are taken from T A (resp. from All these random quantities are employed to construct the capital system (Σ(t) : t ≥ 0) via the graphical representation. Since (P A : A ∈ A k ) is a collection of independent Poisson processes and by definition of the U 's and theŪ 's, the capital system is a stochastic Ising model.
We implies also that J and σ could be considered only in a finite region.
We define the events By the independence properties of the considered random quantities, it results and this gives P F Assume from now on that F M,M occurs.
We claim that Σ u (M ) = +1, for each u ∈ B 4k (v). In particular, on F We also claim that Σ u (M ) = σ u (M ) for each u / ∈ B 4k (v). We prove it by separating two subcases: We now need to check the following: We can define recursively the sequence of random times (ψ : ∈ N). Let for ≥ 1 and the conventional agreement that ψ 0 = M .
For each ≥ 1 one has that the random set has infinite cardinality a.s.
In the interval [M , ψ 1 ) we are dealing only with arrivals having A ∈ A k such that (A ∪ ∂ ext A) ∩ B 4k (v) = ∅, hence belonging to Υ 1 . Therefore, by construction and from the fact that Σ u (M ) = σ u (M ) for each u / ∈ B 4k (v), the original and the capital system share the same flips in Υ 1 . This leads to Now we analyze the arrival in ψ 1 . There exist unique (a.s.) A ∈ A k and n ∈ N such that the arrival τ A,n = ψ 1 . Four cases can be considered for the set A. They are presented and discussed below.
By Lemma 3 we know that ∆ A HJ (Σ(ψ 1 )) ≥ 2. However, the occurrence of F M guarantees that this arrival does not correspond to a flip.
By the selected interactionsJ for the capital system (in particular: if e = {x, y} ∈ E with x ∈ ∂ ext B 4k (v) and y ∈ ∂B 4k (v), then J e = σ x (M ) = σ x (M ) = Σ x (M )) we are in the position of applying Lemma 3. Analogously to the previous case, F M assures that there is not a flip at A in ψ 1 .

Formula (4) says that
The occurrence of F Re-sampling the interactions of the original system only over the finite set there is a positive probability that it coincides with the interactions of the capital system, since α ∈ (0, 1).
Therefore, also in the original one, the probability that |N v | = ∞ is positive. Hence, by ergodicity, ρ I must be larger than zero. This concludes the proof.

We can conclude that for any initial configuration σ the model is of type M or I. This means that there
is not convergence of the process almost surely.
We now present a theorem in which we explore the possibility that ρ I is larger than zero without requiring the positivity of the temperature profile. The basic assumption is that some specific sites fixate.
We address the reader to the next section for sufficient conditions to obtain ρ F > 0.
(ii) For any J coinciding withĴ S on S and for any σ

Then, the model is of type M.
Proof. First of all, we notice that there exists J ∈ {−1, +1} E coinciding withĴ S on S with positive probability, since S is finite and α ∈ (0, 1).
By (36) and by ergodicity one has that ρ F > 0. So, it is sufficient to prove that ρ I > 0.
In the following we use the probability measure P (·) = P(·| e∈S {J e =Ĵ e }).
We define the setÂ By definition ofσ R , we know thatÂ recurs with positive probability. The set D in the statement of the theorem depends on σ. We write for D ∈ A k such that D ⊂ R . Thus,Â = D∈A k :D⊂R A D is a finite union. Therefore, one can select ā D ∈ A k withD ⊂ R such that AD recurs with positive probability. Now, the set B, in the frame of Lemma 2, is taken as where 1/2, in the previous formula, is due to ∆DH J (σ) ≤ 0.
By the independence of the Poisson processes, (37) is larger than or equal to where K is given in (17).
Lemma 2 guarantees that B recurs with positive probability. Thus, the ergodic theorem assures that Example 1. In Figure 1 we provide an example. It is here represented a region of the d-graph G, with d = 2. For the notation, see Theorem 3. Consider k = 1, α ∈ (0, 1), γ ∈ (0, 1] (resp. γ ∈ [0, 1)) and T fast decreasing to zero. Set the black bullets as sites with spin +1 (resp. −1) and the other spins We finish the section with a particular case in which we show that the parameter k plays a role to establish the class of a (k, α, γ; T )-model. It is known that the (1, α, γ; T )-model on the hexagonal lattice is of type F when T ≡ 0 (see [23]) and in the next section this result is proven when T is fast decreasing to zero (see Theorem 5). It is important to notice that the hexagonal lattice in our framework is a 2-graph.
In   We denote with σ(∞) = (σ x (∞) : x ∈ V H ) the random limit configuration. By ergodicity and since γ = 1/2 we obtain that for any v ∈ V H .
As in [23] we define the domain wall D H as a subset of the hexagonal dual lattice, which separates the positive spins from the negative ones by joining the center of the hexagons (see Figure 2). By (38) D H = ∅ almost surely. The dual lattice is formed by three classes of edges: the vertical edges, the ascendent ones and the descendent ones (see Figure 2). As in (38), there exists the density ρ ver (ρ asc , ρ des , resp.) of the vertical (ascendent, descendent, resp.) edges of the domain wall. By symmetry and being D H = ∅, we have that ρ ver = ρ asc = ρ des > 0 almost surely.
On each site v ∈ V H with σ v = −1 we consider the triangle Φ v,H connecting the centers of the three hexagons containing v. Such triangle is a closed convex set, in that it is given by its internal part and its boundary. We introduce the set containing all such triangles The set D H is the boundary of the set Φ H . Simplicial homology arguments (see [27]) gives that the domain wall D H can be decomposed in connected components, each of them belonging to one of the following families: If the density of the angles π/3 is different from zero (see Figure 2, right panel), then we have two edges {x, y}, {y, z} ∈ E H such that the random time

Conditions for ρ F > 0
In this section we give some sufficient conditions to obtain ρ F > 0. Let us consider a d-graph and We are now ready to present the following where d G is the maximal degree of the graph G.
Proof. We rewrite the expression in (39) as The term inside the square brackets belongs to [−d G , d G ]. In fact, for any x ∈ Λ, by definition we have By (40), we have the thesis.
We give two more definitions that will be used in the proof of the next lemma. They are based on the embeddedness of the d-graph and on the concept of cell. For a d-graph G = (V, E) and ∈ N we define We claim that there exists the following bound for H ,J (σ(t)) where kd G represents an upper bound for ∆ A H J (σ), for each σ ∈ {−1, +1} V and A ∈ A k while −2 is a lower bound for the case in favour to the Hamiltonian, and comes out from (2).
Notice that the sets A ∈ A k such that A ∩ [− , ) d = ∅ do not appear in (41) because, in this case, We check (41) by recursion.
By the finiteness of the set of all the A ∈ A k intersecting [− , ) d , then there exists a unique (a.s.) Then Thus, by induction one obtains (41).

By Lemma 4 one has
By applying the expected value operator to the terms in inequality (41) and by (42), one has Now, we bound the single terms of inequality (43).
By (14) in Lemma 1, one has where the constant C 1,k can be chosen in such a way that it does not depend on . In fact, by translation invariance, the expected value over a cell is equal to the expected value on any other cell.
By the bound on the cardinality of Z we obtain Now, by contradiction, suppose that there existsĀ ∈ A k such that E[|S + A,∞ |] = ∞. For large enough, one has By (43)-(46) and for large enough, one gets The monotone convergence theorem leads to lim t→∞ E[|S + A,t |] = ∞. Let us take t = 1/2 . Therefore We notice that only in the case of a temperature profile T that does not satisfy Corollary 1 one can hope to obtain ρ F > 0.In the following we work under the assumption that the temperature profile T is fast decreasing to zero and we give, jointly to it, some sufficient conditions to obtain ρ F > 0.
Next two theorems provide conditions for ρ F > 0. Analogous results are in [14,23] for the case k = 1.
Proof. Let us consider A ∈ A k with A v. As in [14,23], we notice that if t ∈ T A corresponds to a flip at A, then t ∈ S − A,∞ ∪ S + A,∞ . Formula (14) in Lemma 1 implies that |S − A,t | < ∞ and Lemma 5 states that |S + A,t | < ∞, almost surely. Thus, we have |N v | < ∞ almost surely. By ergodicity, ρ F > 0.

Assume that there exists a finite set B ⊂ V such that |B| > k and for any
Then ρ F > 0. In particular, P( u∈B {lim t→∞ σ u (t) = +1}) > 0.
Proof. Let us consider B as in the statement of the theorem. Since α > 0 and E(B) is finite, then with positive probability one has that J e = +1 for any e ∈ E(B). In fact, the independence of the interaction Hence, inequality (48) and conditions σ v = +1 for each v ∈ B and J e = +1 for each e ∈ E(B) imply By formula (16) in Lemma 1, there is a positive probability that all the spins maintain the initial value +1 in the region B. By ergodicity, we obtain ρ F > 0.
We present here a 2-graph on which Theorem 5 can be applied when k = 2 (see Figure 3) is v. Each connected set A v of cardinality 2 is such that |Γ A | = 5. The case A = {v} is such that For what concerns Theorem 6, we address the reader to Figure 1. In this case we consider k = 1.

Take B as the set of the black bullets. A is a set containing only a vertex of B. It is clear that Theorem
6 can be applied.

Remark 5.
(i) For any given J , ifσ C is k-absent on J , then it is also k -absent on J , for each recurs with null probability by using the probability measure Proof. We consider the Harris' graphical model and denote it by Z = (Z t : t ≥ 0). By hypothesis, J of the model coincides withĴ over the setÊ(C).
By contradiction, assume that A(σ C ) defined in (49) recurs with positive probability.
Define the random set Therefore K is a function of the process Z. Define the set B = (Z t : t ∈ [0, 1]) : K = {t 1 , . . . , t m } with 0 < t 1 < · · · < t m < 1 and where m and the A (j) 's are given as in Definition 4.
We notice that we are under the conditions of Lemma 2. In fact, because the probability that the U 's are smaller than 1/2 and that the only finite sequence of arrivals in [0, 1] of the Poisson processes P A , with A ∈ A k and A ∩ C = ∅, is ordered as in K, is larger than zero (see the proof of Theorem 3 for a similar argument).
where P is the conditioned probability as in (50), and ρ F > 0.
Proof. We only prove the case when any σ C ∈ {−1, +1} C , with σ u = σ v , is k-absent onĴ . The other case is analogous.
Let us suppose that J coincides withĴ onÊ(C). The vertices u and v, as defined in the theorem, flip simultaneously with probability zero because ν G (u, v) ≥ k. In fact, for any A ∈ A k , the set {u, v} is not a subset of A. By Theorem 7, we know that the set with zero probability. Therefore there exists lim t→∞ σ u (t), lim t→∞ σ v (t) P -a.s., and they have opposite sign P -a.s..
So, if we remove the conditioning assumption that J coincides withĴ onÊ(C), we have that the spins on the vertices u, v fixate with positive probability.
By ergodicity we obtain that ρ F > 0.
We present an example in which Theorem 8 can be applied.
Example 3. We consider k ≤ 3 and a 1-graph G = (V, E), as in Figure 4, whose cell has vertices (the cell will be scaled by 1 5 and translated by −  We now distinguish the cases of σ 1 = +1 and σ 1 = −1. Consider σ 1 = +1. Therefore lim t→∞ σ 6 (t) = lim t→∞ σ 11 (t) and this also implies that lim t→∞ σ i (t) = lim t→∞ σ 6 (t), for i = 7, 8, 9, 10, 11. We notice that all Two important remarks are in order: first, the model described in this example is of type M; second, this model can be extended to d-dimensional graphs (see Figure 5).
The authors show that the model is of type I. With a coupling argument one can prove that also for J ≡ −1, i.e. α = 0, the system is of type I (see [14]). In fact, consider two systems on the d-graph E d ), γ = 1/2 and J ≡ +1 for one of the systems andJ ≡ −1 for the other one. We say that We take a initial configuration σ for the system with J and initial configurationσ for the system withJ as follows Therefore, also the random vector Therefore the two systems are of the same type (I, F or M). The same construction works for any bipartite graph.
Example 3 suggests a general result on a particular class of graphs. We firstly define such a class, and then present a simple and meaningful theorem. The proof of such a result can be viewed as a generalization of the arguments carried out in Example 3. (i) two adjacent cycles have only one vertex in common. We denote such vertices as common vertices, and collect them in the set V I .
The vertices V ⊂ V Γ will be called original vertices.
Notice that V ⊂ V I . In Figure 5 we have the representation of Γ 3,3 (L 2 ). Proof. We briefly denote ν Γ ,m (G) by ν. Consider two original vertices u, v ∈ V such that ν(u, v) = m.
The finite subgraph composed by the m cycles between u and v will be denoted by The cycles composing Γ u,v are subgraphs as well. They will be denoted by , so that if one takes h, = 1, . . . , m such that h < , then for each x ∈ V h and y ∈ V it results ν(x, u) < ν(y, u). We consider the first three cycles Y 1 , Y 2 and Y 3 and we label the vertices of these cycles in the following way Let us consider the following condition With positive probability, there exist interactions J that satisfies the previous condition (H1). Define We prove that the configurations belonging to are 1-absent on the interactions J that satisfy condition (H1). Without loss of generality in (51) one can take σ v 1 = +1 and σ v 2 = −1. Now we consider two cases (i) σ v1 = σ v3 = +1 and σ v2 = −1; In case (i) let us consider the following elementŝ In case (ii), we defineÃ For σ C = (σ v : v ∈ C) which satisfies item (ii) we eliminate from the sequence (Â Since ν(v 1 , v 2 ) = ≥ k, then Theorem 8 guarantees that, for large t, σ v 1 (t) and σ v 2 (t) fixate and are equal. Therefore, ρ F > 0.
With an analogous argument one can prove the same result for v 3 and v 4 , i.e. there exist the limits lim t→∞ σ v3 (t) and lim t→∞ σ v4 (t), and they coincide.
We analyze separately the four cases.
We recall the definition of cluster that will be used in the next Theorem. Let us consider a d-graph For a vertex x / ∈ Z 2 let us take the two vertices u, v ∈ Z 2 having Euclidean distance equal to one and such that ν(x, u), ν(x, v) < ν (u, v). The fact that p = 0 means that the event recurs with probability one. By symmetry of the graph, the events E 1 (t), E 2 (t), E 3 (t), E 4 (t) have the same probability. Therefore, using also that γ = 1/2, one obtain lim inf t→∞ (E i (t)) ≥ 1 8 , ∀ i = 1, 2, 3, 4.
We notice that E i (t) is an increasing event, for each i = 1, 2, 3, 4.
Let us define By the fact that the events E i 's are increasing and k = 1 one can use the FKG inequality to bound the probability of E(t) (see e.g. [18]). Therefore lim inf t→∞ P(E(t)) ≥ lim inf t→∞ (P(E N (t))P(E E (t))P(E S (t))P(E O (t))) ≥ 1 8 In fact, if the set of vertices with negative spins is not empty, then there exists at least one vertex that can change its spin from −1 to +1 with a flip that is indifferent for or in favour of the Hamiltonian.
Then, recursively, all the spins of the vertices belonging to [−L, L] 2 can become positive at a same time.
By Lemma 2, one obtains that B L recurs with probability larger or equal than 1 8 4 . Therefore, for each L ∈ N, the probability that there are not vertices belonging to [−L, L] 2 whose spins fixate to −1 is at least 1 8 4 . This last assertion contradicts formula (53) when ε < 1 8 4 .

Conclusions
In this paper we have presented a generalization of the Glauber dynamics of the Ising model. The temperature is assumed to be time-dependent and fast decreasing to zero, hence including the case of T ≡ 0. Moreover, it is allowed that spins flip simultaneously when belonging to some connected regions.
The dynamics is taken over general periodic graphs embedded in R d . The obtained results can be compared with the standard case of zero-temperature, cubic lattice and k = 1.
For the cubic lattice L 2 the paper [23] says that for α = 1 or α = 0 and γ = 1 2 the model is of type I. On the same graph, [14] proves that the model is of type M when α ∈ (0, 1) and γ = 1 2 (actually, their arguments hold true for γ ∈ (0, 1) as well). In [13,22] it is shown that, for L d with d ≥ 2, the model is of type F when α = 1 and γ sufficiently close to one (or, by symmetry, sufficiently close to zero). In particular, it is known that the limit configuration is given by spins whose values are +1 (or, by symmetry, −1). For a better visualization of the results, see Figure 6 (left panel).
When T is fast decreasing to zero and positive, and by considering L d with d ≥ 2, then it is possible to exclude that the (k, α, γ; T )-model is of type F (see Theorem 2) and refer to Figure 6 (right panel).
We feel that Figure 6 might also contribute to highlight some problems left open by this paper.
To conclude, we think that our results may represent a first move towards the following three conjectures.
(i) If α, γ ∈ (0, 1) and T is fast decreasing to zero, then the type of the (k, α, γ; T )-model over a d-graph G = (V, E) is identified by the value of k ∈ N and by the graph G.