STOCHASTIC ISING MODEL WITH FLIPPING SETS

STOCHASTIC ISING MODEL WITH FLIPPING SETS OF SPINS AND FAST DECREASING TEMPERATURE ROY CERQUETI AND EMILIO DE SANTIS Abstract. 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 Rd . The interactions between couples of spins are assumed to be fixed or i.i.d. 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. By adopting the classification proposed in [14], we present conditions in order to have models of type F (any spin flips finitely many times), I (any spin flips infinitely many times) and M (the mixed case). Several examples are provided in all dimensions and for different cases of graphs. The most part of the obtained results holds true also for the case of zero-temperature and some of them for the cubic lattice Ld = (Zd , Ed ).

1. 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. [19], 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 respect to the Gibbs measure of the Ising model. The Markov process describes the stochastic evolution of spins, which are binary variables ±1 on the vertices of an infinite periodic graph G = (V, E) with finite degree. Such class of graphs includes the meaningful standard case of d-dimensional cubic lattices Ld = (Zd , Ed ), for d ∈ N. The interactions J = (Je : e ∈ E) are fixed or i.i.d. random variables having Bernoulli distribution µJ (Je = +1) = α and µJ (Je = −1) = 1 − α, with α ∈ [0, 1], and they 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 B(R) and such that T (t) is the temperature at time t for any t ∈ [0, +∞). In most of the results it is assumed that limt→+∞ T (t) = 0, and sometimes the positivity is required. Measurability is not always a required assumption on T , and some results hold true also for T not measurable. The case of T not constant leads to a non homogeneous Markov process. In this context, the most important setting will be the fast decreasing temperature profile, defined below, due to its connections with the well studied case of T ≡ 0. The initial configuration of the Markov process is σ(0) ∈ {−1, +1}V . Its components are assumed to be fixed or i.i.d. and randomly selected by a Bernoulli measure νσ(0) with parameter β ∈ [0, 1]. When the interactions and the initial configuration are random, the considered stochastic Ising models are denoted as (k, α, β; T )-model on a periodic graph G. The dynamics will be specified in more details in Section 2. 1

2

ROY CERQUETI AND EMILIO DE SANTIS

The question we deal with is: (Q) Does a given spin on v ∈ V flip infinitely many times a.s.? This problem is paradigmatic in the literature studying zero-temperature stochastic Ising models (see e.g. [2, 3, 10, 11, 13, 14, 23, 4, 5, 17, 12]). Indeed the results that we will obtain, with the exception of Theorem 4, hold true also for the case of T ≡ 0. In the frame of random interactions the question (Q) should be rephrased. In fact it is not always meaningful to deal with single sites, because the random environment leads to sites differently behaving even if the graph is periodic. In the important 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. On this basis we obtain 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, β, T . Indeed, the main part of our results holds true for α ∈ (0, 1) and β ∈ [0, 1] under some natural requirements on the decay rate of the temperature profile T . We now mention some papers, which are particularly close to our study. In a very different context, [1] has shown the recurrence of annihilating random walks on Zd under very general conditions. This work has as a consequence that the one-dimensional stochastic Ising model with α = 1, β ∈ (0, 1) and T ≡ 0 is of type I. In [14] there is an analysis of the zero-temperature case for the cubic lattice Ld = (Zd , Ed ), β = 1/2 and different product measures µJ over R for the interactions J . Among the results, the authors provide a complete characterization for L1 and for all the measures µJ , i.e. they identify the classes I, F, and M for each µJ . In doing so, [14] adapts and extends [1] in this specific context. Moreover, the authors identify the class M when d = 2 for the cubic lattice and the measure µJ is a product of Bernoulli type with parameter α ∈ (0, 1). In [24] there is a treatment of the case with α = 1 (homogeneous ferromagnetic case) and β = 1/2 for L2 , where it is proven that the model is of type I. Under the same conditions, [2] refines [24] in discussing the recurrence and the growth of some geometrical structures. It is also worth noting that [24] analyzes the framework with µJ continuous 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, Ld and β > βd? ' 1. The authors show that all the spins converge to +1, hence leading to a model of type F. The paper [23] extends [13] and shows that limd→+∞ βd? = 1/2. In this last context, to prove that for a large density of initial plus the process converges to +1, it is worth mentioning [3] in which the stochastic Ising model at zero-temperature is analyzed on a d-ary regular tree Td . Analogously to [13, 23], the authors prove that for β > βˆd , where βˆd is in (0, 1), all the spins converge to +1. Moreover it is also shown, along with many other results, that limd→+∞ βˆd = 1/2. For constant and positive temperature T question (Q) does not make sense because all the spins flips infinitely many time, but there are related results for the convergence of the system to the Gibbs state in the case of low temperature (see e.g. [6, 7, 9, 21]) or high temperature (see e.g. [8, 6, 22, 20]). In particular, in [6] it is shown a different approach to the equilibrium Gibbs measure in relation of the temperature in a regime of random interactions (random environment). We

ISING MODEL WITH FLIPPING SETS AND FAST DECREASING TEMPERATURE

3

believe that there is a connection between the behavior of the stochastic Ising model with α ∈ (0, 1) at low, constant and positive temperature and at zero-temperature dynamics. In this respect, it seems that the type of the model, F or M, at zerotemperature is related to properties of metastability of the same model at low constant temperature. The geometric structure of the support of the spins is also relevant in both the cases of zero- and low constant temperature. In [14, 6] some of the main results come out from a combinatorial-geometric interpretation of the underlying graphs. For cases different from Ld at zero-temperature, we mention [3] and [12], with tree-related graphs, and [17], where the hexagonal lattice is considered. In this last paper the authors moves from [24], where it is proven that sites fixate, and show that the expected value of the cardinality of the cluster containing the origin is infinite. As already preannounced above, we aim at providing an answer to (Q) in the framework of very general graphs. In particular, we deal with graphs which are periodic and embedded in Rd . After providing a definition of the dimension of such graphs, we provide results over all the dimensions. In doing this, we advance the previous literature, which is mainly focussed on L1 (see e.g. [14]), L2 (see e.g. [14, 24, 2]) and the hexagonal lattice (see e.g. [17]) with the noticeable exceptions of the general dimensional cubic lattices Ld in [13, 23]. It is important to note that this topic is important either at a purely theoretical level as well as in the applied science. Indeed, we mention [25, 10, 29, 28], 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]. Our results aim to generalize those related to the stochastic Ising models at zero-temperature over the cubic lattice Ld . (i) We allow for simultaneous flips of the spins, hence leading to flipping regions rather than the single spin. 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 a single individual). The cardinality of the flipping sets is constrained by a parameter k, which will be formalized below. (ii) The considered graphs are periodic, infinite and with finite degree. This framework includes naturally Ld , for each d ∈ N. This generalization allows us to provide theoretical results and meaningful and physically consistent examples (like crystal lattices) outside the restrictive world of Ld . (iii) The temperature is taken not necessarily zero. Specifically, we consider a fast decreasing to zero temperature and requires 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. It is also worth stressing that all the results where the temperature is not positive hold true also in the zero-temperature setting. We now list and comment our results. Theorem 1 gives sufficient conditions for the temperature profile having a similar or different behavior of the zero-temperature case. It is shown that the model is of type I when the temperature does not decrease fast to zero, whereas a spin changes on a single site a finite number of times (a.s.) in the converse case. This result depends on the Hamiltonian, and can then be easily rewritten for Gibbs models endowed with a Glauber-type dynamics.

4


Some conditions of Theorem 1 can be replaced with weaker ones over Ld or over graphs with even-degree sites. In Theorem 4 we present a result stating that the model is of type M or I, under some conditions. Specifically, it is required that the temperature profile T is fast decreasing to zero and positive and the graph belong to the rather wide class of stable d-Egraphs (see next section for the formal definition of this concept). Such class contains also all the cubic lattices Ld . Theorem 5 states some conditions to obtain models of type M. Results therein contained hold true also in the frame of T ≡ 0 and are connected with the findings in Section 4. In Section 4 we do not longer use the positivity of the temperature. Therefore all the results contained in such section are valid for the particular case T ≡ 0, and describe conditions leading to fixating sites. Item a. of Theorem 7 is inspired by [24, 14] 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. Item b. of Theorem 7 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. This is grounded on the positive probability that the sites will be all equal and they will have all positive interactions. In Definition 3 we adapt to our setting the concept of e-absent configurations (namely, k-absent here), which has been introduced for the graph L2 in [2, 14]. In details we include k > 1 and the considered class of graphs. Theorems 8 and 9 are based on k-absence. The former result states that k-absent configurations can appear only on a random finite time interval (a.s.); 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, then it is (k + 1)-absent. Hence, a large value of k seems to facilitate the fixation of the sites (see the enunciation of Theorem 8). Differently, a large value of k is an obstacle for the fixation of the sites in Theorem 9 (see the enunciation). 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 proposed a result in which, under some conditions, the model is of type F for k = 1 and it is not of type F for k ≥ 2 (see Theorem 6). The provided examples illustrate a wide part of the outcomes, and complement the theoretical findings of the paper. In particular, we have introduced Γ`,m (G), which can be viewed as graphs constructed by replacing the original edges of a graph G = (V, E) with more complex structures (see Definition 4). For this specific class, we have provided some conditions for which the models over such graphs are of type M (see Theorem 11), I (see Theorem 12) and F (see Theorem 13). The last section of the paper concludes and offers some conjectures and open problems. In order to assist the reader, we have provided a graphical representation of 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 stable d-Egraphs at positive temperature (see Figure 6). 2. Definitions and first properties of the model In this section we will define a dynamical stochastic Ising model and the main notations used throughout the paper. 2.1. Graphs. For our results we consider different kinds of graphs but in all the analyzed cases they will have finite maximal degree.


5

We define a graph G = (V, E) embedded in the space Rd , i.e. the vertices are points and the edges are lines. The degree of a vertex v ∈ V is denoted by dv and by hypothesis it is finite. Moreover all the considered graphs are translation invariant in a sense that will be specified below. For such a graph G = (V, E), we also say that the dimension d is minimal if there exists a base {v1 , . . . , vd } of Rd such that the graph is translation invariant with respect to the translations vi ∈ Rd , for i = 1, . . . , d. In the sequel we always consider the graphs embedded in Rd where the dimension d is minimal. We also suppose that in any finite subset C ⊂ Rd in the embedding there are only a finite number of vertices. The graphs with all the previous properties will be denoted d-graphs, and they are widely used in the field of physics for modeling crystal structures. Using an affine transformation we can always consider that a d-graph is invariant by translations of ei with i = 1, . . . , d, where {e1 , . . . , ed } is the canonical base of Rd . In the following we adopt this point of view by considering that the d-graph G is translation invariant with respect to the canonical base. Therefore we can construct a d-graph G = (V, E) doing a tessellation of Rd with the basic cell Cell = [0, 1)d , i.e. the space Rd is seen as the union of disjoint hypercubes [0, 1)d + z with z ∈ Zd . It is clear that a d-graph G = (V, E) can be construct in a way that there is no vertices of G on the boundary of Cell. To specify the d-graph G = (V, E) it is enough to give the vertices VCell inside Cell with the edges ECell = {{x, y} ∈ E : {x, y} ∩ Cell 6= ∅}. We denote with dG = sup{dv : v ∈ V } the maximal degree of the graph G = (V, E); by periodicity and the fact that Cell contains a finite number of vertices, then dG is finite for any d-graph G. Moreover if a d-graph has at least a vertex with even degree (therefore infinite such vertices due to the translation invariant property) we denote it as d-Egraph. As an example of a d-Egraph we take the lattice Ld = (Zd , Ed ) (see [15]), where the degree of any vertex is equal to 2d and it can be embedded in Rd in a natural way. Now we introduce a new condition on the d-Egraphs. Definition 1. We say that a d-Egraph G = (V, E) is stable if there exists a vertex v ∈ V having dv even with the following property. For any finite A ⊂ V with |A| > 1, v ∈ A and such that there exists y 6∈ A with {v, y} ∈ E, one has: |{e = {x, y} ∈ E : {x, y} ∩ A 6= ∅}| − dv > 0. An easy example of a stable d-Egraph is the cubic lattice Ld = (Zd , Ed ). Given n ∈ N, a path γ of length n starting in x ∈ V and ending in y ∈ V is a sequence of vertices (x0 = x, x1 , . . . , xn−1 , xn = y) having {xi−1 , xi } ∈ E for each i = 1, . . . , n. A set A ⊂ V is connected if for any couple x, y ∈ A there exists a path γ = (x0 = x, x1 , . . . , xn−1 , xn = y) with xi ∈ A, for i = 0, . . . , n. We say that a graph G = (V, E) is connected if V is connected. The distance νG (u, v) in G of two vertices u, v ∈ V is the length of a shortest path starting in u and ending in v. For u ∈ V and L ∈ N we define the ball centered in u with radius L as BL (u) = {v ∈ V : νG (u, v) ≤ L}. 2.2. Hamiltonian. For a given σ ∈ {−1, +1}V and for any subset A of V , we (A) write σ (A) = (σv : v ∈ V ) to denote the configuration having ½ −σy , if y ∈ A; (A) σy = (1) σy , if y 6∈ A;

6


that corresponds to flip the configuration σ on the set A. The formal Hamiltonian associated to the interactions J is X HJ (σ) = − Je σx σy .

(2)

e={x,y}∈E

However, the definition (2) is not well posed for infinite graphs. We will work with the following: the increment of the Hamiltonian at A, for a finite set A ⊂ V , is X X ∆A HJ (σ) = −2 Je σx(A) σy(A) = 2 Je σx σy . (3) e={x,y}∈E:x∈A,y6∈A

e={x,y}∈E:x∈A,y6∈A

In the following, using (3), we will construct an infinitesimal generator that define a Markov process. It is worth noting that the value of ∆A HJ (σ) can be only an even integer. Therefore, if ∆A HJ (σ) > 0, then ∆A HJ (σ) ≥ 2 and, analogously, ∆A HJ (σ) < 0 means that ∆A HJ (σ) ≤ −2. 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. The process is denoted by σ(t) = (σv (t) : v ∈ V ) ∈ {−1, +1}V and it is taken with left continuous trajectories. We call Ak the collection of the connected subsets A ⊂ V having cardinality smaller or equal to k ∈ N. It is important to note that, for a given v ∈ V , the set {A ∈ Ak : v ∈ A} is finite. Therefore, the set Ak is countable and, following [19], we can define the infinitesimal generator related to Ak as follows: X J ,(A) Lk,J (f (σ)) = ct (σ)(f (σ (A) ) − f (σ)), (4) t A∈Ak

where, for t ≥ 0 such that T (t) > 0, e−∆A HJ (σ)/T (t) 1 = . ∆ H (σ)/T (t) −∆ H (σ)/T (t) 2∆ H A J A J A J (σ)/T (t) e +e 1+e In the case of T (t) = 0, for t ≥ 0, we define J ,(A)

ct

(σ) =

J ,(A)

ct

(σ) = lim+ T →0

e−∆A HJ (σ)/T . + e−∆A HJ (σ)/T

e∆A HJ (σ)/T

(5)

(6)

Hence, when T ≡ 0, we have: J ,(A)

• if ∆A HJ (σ) > 0 then ct (σ) = 0; J ,(A) • if ∆A HJ (σ) = 0 then ct (σ) = 12 ; J ,(A) (σ) = 1. • if ∆A HJ (σ) < 0 then ct We remark that in the case of T ≡ 0 and k = 1 our dynamics is the Glauber dynamics at zero temperature; it is a well developed field of the statistical mechanics and of physics (see [24, 23, 14, 2, 11] and citations therein). The previous defined process can be construct using a collection of independent Poisson processes (PA : A ∈ Ak ) with rate 1, in the so called Harris’ graphical representation [16]. Call TA = (τA,n : n ∈ N) the arrivals of Poisson process PA . Conditioning on the event {t ∈ TA } the probability that there is a flip on the set A, i.e. σ(t+ ) = σ (A) (t), J ,(A) is equal to ct (σ(t)). An useful representation of such probabilities can be given by considering the family (UA,n : A ∈ Ak , n ∈ N)

(7)


7

J ,(A)

of uniform i.i.d. random variables such that: if UA,n < cτA,n (σ(τA,n )), then there is a flip of the set A at time τA,n (see [16, 19]). The representation of the Markov process based on the Poisson processes is very popular in the framework of zero-temperature, since it exhibits remarkable advantages with respect to the one based on the generator. First, the spatial ergodicity of the process with respect to the translations of the e’s at a given time t can be invoked on the ground of a very general theory (see [16, 18, 24, 26]). Second, Poisson processes-based representation is the natural setting for the proofs of the results. We say that a flip at A at time t ∈ TA is in favor of the hamiltonian if ∆A HJ (σ(t)) < 0, it is indifferent for the hamiltonian if ∆A HJ (σ(t)) = 0 and it is in opposition of the hamiltonian if ∆A HJ (σ(t)) > 0. Hence, in the case of a flip in A J ,(A)

• in favor of the hamiltonian, then ct is larger than 1/2; J ,(A) • indifferent for the hamiltonian, then ct is equal to 1/2; J ,(A) • in opposition of the hamiltonian, then ct is smaller than 1/2. We define, for any x ∈ V , [ = {t ∈ TA : at time t there is a flip in opposition of the Hamiltonian}

Nx−

A∈Ak :A3x

the set of flips in opposition of the Hamiltonian involving the site x. Moreover, we define the set of total flips involving the site x as [ Nx = {t ∈ TA : at time t there is a flip in A}, A∈Ak :A3x

and the set of total arrivals involving the site x as [ Qx = {t ∈ TA }. A∈Ak :A3x

We now provide the definition of the probability measure associated to the dynamical model defined in (4), which can be written as: Pk,µJ ,νσ(0) ;T = µJ × νσ(0) × Pk,T ,

(8)

where µJ is the probability distribution of the interactions J forming the random environment; νσ(0) is the probability distribution of the initial configuration σ(0); Pk,T is the measure of the arrivals of the i.i.d. Poisson processes on the sets A ∈ Ak and the sequences of independent U ’s as in (7) – independent also from the Poisson processes – where T is the temperature profile and k ≥ 1. Some particular cases are important in our context. The considered measures over the interactions are of two families: Y µJ = Bere (α), α ∈ [0, 1], (9) e∈E

where Bere (α) is the Bernoulli distribution with parameter α and support {−1, +1}, labeled by the edge e ∈ E. The deterministic case is µJ = δJ ,

J ∈ {−1, +1}E .

(10)

Measure µJ in (9) is the case of ±J model in [14], while (10) is the case of deterministically fixing the interactions on the edges of the graph. Analogously, for the initial configuration σ(0), we use: Y νσ(0) = Berv (β), β ∈ [0, 1]; (11) v∈V

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νσ(0) = δσ , σ ∈ {−1, +1}V . (12) To avoid a cumbersome notation, we denote Pk,µJ ,νσ(0) ;T as: • Pk,α,β;T when µJ and νσ(0) are as in (9) and (11), respectively. Accordingly, the process will be denoted as (k, α, β; T )-model. • Pk,hJ i,β;T when µJ and νσ(0) are as in (10) and (11), respectively. The process will be denoted as (k, hJ i, β; T )-model. • Pk,α,hσi;T when µJ and νσ(0) are as in (9) and (12), respectively. The process will be denoted as (k, α, hσi; T )-model. • Pk,hJ i,hσi;T when µJ and νσ(0) are as in (10) and (12), respectively. The process will be denoted as (k, hJ i, hσi; T )-model. Remark 1. To fix the interactions J and/or the initial configuration σ(0) (hence, leading to the indices hJ i and hσi, respectively) has an important role in the proofs of some results, as we will see below. In fact, in our cases, when a property is shown for all fixed J and/or σ(0), then such a property holds true also for a general Bernoulli µJ and/or νσ(0) . The definition of the measure associated to the model allows us to introduce the densities of interest in assessing the type of the models. A vertex x ∈ V is said of kind finitely flipping (resp. infinitely flipping) if Nx < ∞ (resp. Nx = ∞) Pk,α,β;T -almost surely. Using standard ergodic arguments one can see that for a (k, α, β; T )- model the quantity ρI = ρI (k, α, β; T ) = lim

k→∞

|{x ∈ Bk (v) : x flips infinitely many times}| |Bk (v)|

(13)

does exist Pk,α,β;T -almost surely, and it does not depend on the vertex v (for details in the case of the cubic lattice see [24, 14]). We also define |{x ∈ Bk (v) : xflips finitely many times}| ρF = ρF (k, α, β; T ) = lim . k→∞ |Bk (v)| Therefore ρI + ρF = 1. By adopting the notation of [14], we can say that: • If ρI = 1 (ρF = 1, resp.) we write that the (k, α, β; T )-model is of type I (F, resp.). • If 0 < ρI , ρF < 1 we write that the (k, α, β; T )-model is of type M. 3. Conditions for ρI > 0 This first result is given in a general setting, allowing to fix the initial configuration σ(0) and the interactions J ; moreover it could be presented for a completely general Glauber dynamics associated to a Gibbs measure and, thus, associated to a Hamiltonian. Theorem 1. Consider a connected d-graph G = (V, E). For any interaction J ∈ {−1, +1}E , initial configuration σ ∈ {−1, +1}V , d, k ∈ N and temperature profile T , the (k, hJ i, hσi; T )-model is such that: • If lim sup T (t) ln t < 4 (14) t→∞

then, for any x ∈ V , Nx− is finite Pk,hJ i,hσi;T -almost surely. • If, for a given x ∈ V , lim inf T (t) ln t > 4dx t→∞

then Nx is infinite Pk,hJ i,hσi;T -almost surely.

(15)


9

Proof. We start by proving the first item of the theorem. By (14), for x ∈ V , we can write ρ=

1 · lim sup T (t) ln t < 1. 4 t→∞

(16)

We also fix a constant α ∈ (1, ρ1 ), hence ρ < α1 . Given Nx− (resp. Nx ), for x ∈ V − and n ∈ N, we define Nx,n = Nx− ∩ [n − 1, n) (resp. Nx,n = Nx ∩ [n − 1, n)). We prove that Z E(|Nx− |) =

Ω

|Nx− (ω)|dPk,hJ i,hσi;T (ω)

− is finite obtaining that the set Nx,n is Pk,hJ i,hσi;T -almost surely finite. We observe that there exists n ¯ ∈ N large enough such that, for any n ≥ n ¯ and for any t ∈ [n − 1, n), J ∈ {−1, +1}E , σ ∈ {−1, +1}V ½ ¾ 1 1 J ,(A) J ,(A) ≥ sup c (σ) : c (σ) < A 3 x. (17) t t (n − 1)α 2 1 In fact, for any t ∈ [n − 1, n) using (5) and the fact that 1+e x decreases in x, one has: 1 1 J ,(A) J ,(A) ct (σ) < ⇒ ct (σ) < . (18) 2 1 + e4/T (t) Hence, for a given ρ0 ∈ (ρ, 1) there exists t0 such that for any t > t0 , by (16), we have 4/T (t) > ln t/ρ0 . Now, it is sufficient to take n = dt0 e + 1. The second inequality in (18) gives that, for any n ≥ n ¯ and for any t ∈ [n − 1, n): J ,(A)

ct

(σ)
1. 4dx t→∞

(21)

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For x ∈ V and n ∈ N, we define the event Fx,n = {τ{x},` ∈ [n − 1, n) for some `}, 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 (Fx,n : x ∈ V, n ∈ N). In particular, the quantity Pk,hJ i,hσi;T (Fx,1 ) is constantly equals to 1 − e−1 . There exists n ¯ ∈ N large enough such that, for any n ≥ n ¯ and for any t ∈ [n − 1, n), J ∈ {−1, +1}E , σ ∈ {−1, +1}V n o 1 J ,(A) ≤ inf ct (σ) A 3 x. (22) n In fact, by (5), for any t ∈ [n − 1, n) one has J ,(A)

ct

(σ) ≥

1 1+

e4dx /T (t)

.

(23)

Therefore, by (21), there exists t0 such that for t > t0 one has 4dx /T (t) < ln t. Analogously to the previous item, define n = dt0 e + 1. Hence, for any n ≥ n ¯ and for any t ∈ [n − 1, n): J ,(A)

ct

(σ) >

1 1 1 = ≥ . ln t 1+e 1+t 1+n

Then by Levy’s conditional form of the Borel-Cantelli lemma (see [30]), one obtains the proof of the statement. ¤ Remark 2. Consider the d-graph Ld = (Zd , Ed ). In this case dx = 2d, for any x. Moreover, fixed a finite subset A ⊂ Zd , then |{e = {x, y} ∈ Ed : x ∈ A, y 6∈ A}| is even. In this case, condition (14) can be replaced with a less restrictive one, as follows: lim sup T (t) ln t < 8. (24) t→∞

E

In fact, for each J ∈ {−1, +1} and σ ∈ {−1, +1}V , the flips in opposition of the Hamiltonian have ∆A HJ (σ) ≥ 4. This statement can be generalized to the case of a site x with even degree dx . In particular, d = 1 is associated to dx = 2, for each x, and also |{e = {x, y} ∈ E1 : x ∈ A, y 6∈ A}| = 2. If there exists limt→∞ T (t) ln t 6= 8, then one among hypotheses (24) and (15) is verified. As an immediate consequence of Theorem 1 we obtain Corollary 2. For any interaction J ∈ {−1, +1}E , initial configuration σ ∈ {−1, +1}V , for d, k ∈ N and temperature profile T , the (k, hJ i, hσi; T )-model, defined on a connected d-graph G = (V, E) is such that: if lim inf t→∞ T (t) ln t > 4dG then, for any vertex v ∈ V , Nv is infinite Pk,hJ i,hσi;T almost surely. Hence, the k, hJ i, hσi; T -model is of type I. The framework introduced above leads to the following: Definition 2. The temperature profile T : [0, +∞) → [0, +∞] is said to be fast decreasing to zero if, for any k ≥ 1, J ∈ {−1, +1}E and σ ∈ {−1, +1}V , there is only a finite number of flips in opposition of the hamiltonian at any site Pk,hJ i,hσi;T almost surely, i.e. for any x ∈ V the set Nx− is finite Pk,hJ i,hσi;T -almost surely. In Theorem 1, first item, we have obtained a sufficient condition, not dependent on the connected d-graph G, to have a temperature profile fast decreasing to zero. We now show that if T is fast decreasing to zero, then Nx− can be empty, for any x ∈ V .


11

Corollary 3. Under the condition that T is fast decreasing to zero, for any interaction J ∈ {−1, +1}E , initial configuration σ ∈ {−1, +1}V , for d, k ∈ N, the (k, hJ i, hσi; T )-model, defined on a connected d-graph G = (V, E), is such that: for any finite set V0 ⊂ V , one has ¡ ¢ Pk,hJ i,hσi;T {|Nx− | = 0 : x ∈ V0 } > 0. (25) Proof. There exists tˆ such that ¡ ¢ Pk,hJ i,hσi;T {|Nx− ∩ [tˆ, ∞)| = 0 : x ∈ V0 } > 0

(26)

for continuity of the probability measure. The probability that all the flips Nx ∩[0, tˆ) are not in opposition of the Hamiltonian is positive. We conclude the proof by noticing that the probability in (25) can be bounded from below by the product of these two positive probability in view of the independence of the U ’s of (7). This conclusion can be obtained also by using the strong Markov property. ¤ Theorem 4. Under the condition that T is fast decreasing to zero and positive, for any α ∈ (0, 1), for any β ∈ [0, 1] and for d, k ∈ N, then the (k, α, β; T )-model, defined on a connected stable d-Egraph G = (V, E), is such that ρI > 0. Proof. By contradiction we suppose that ρI = 0. By the assumption that the dEgraph is stable, we can choose a vertex v ∈ V with even degree satisfying Definition 1. Let us consider the sets B2k (v) and B4k (v), where we are using the constant k given in the statement of the theorem. By hypothesis that ρI = 0, there exists a constant M > 0 such that Pk,α,β;T (TB4k (v)\B2k (v) < M ) > 0,

(27)

where TV0 = inf{t ∈ R+ : ∀t0 ≥ t, ∀x ∈ V0 , σx (t0 ) = σx (t)}, for any finite subset V0 ⊂ V . We notice that TV0 is not a stopping time. Now, consider all the finite possible interactions J1 , . . . , Jq , . . . , JQ ∈ {−1, +1}E obtained by taking all the values of Je ’s for any e = {u, w} having {u, w}∩B2k (v) 6= ∅ and the same value of Je ’s, of the original system, for any e = {u, w} having {u, w} ∩ B2k (v) = ∅, i.e. at least one vertex of e is in B2k (v). The value Q is finite because we are dealing with d-graphs, and depends on G, k and v. We will construct a coupling among all the different systems with interactions J1 , . . . , Jq , . . . , JQ . In any system described above, we take the same initial configuration. Moreover we also take the same collection of Poisson processes (PA : A ∈ Ak ), therefore all the systems can have flips of a set A simultaneously. 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). We set q = 1 for the original system, so that J1 = J and σ (1) = σ. Consider the q-th system, with q = 1, . . . , Q. By using the variables U ’s in (7), which are assumed to be common for all the Q systems, we have that the flip of the Jq ,(A) (q) Jq ,(A) (q) set A, having probability cτA,n (σ (τA,n )), occurs if UA,n < cτA,n (σ (τA,n )), where the U ’s are independent and uniform and are also independent from all the other random variables. Moving from this construction, we obtain that the probability that all these systems are equal until time M is positive, as a consequence of the positivity of T (see (5)), the finiteness of the set B2k (v) and of M . We call FM the event that all the systems remain equal until time M . We have Pk,α,β;T (FM |TB4k (v) < M ) > 0. In fact, let us define Jq ,(A) q ,(A) ηM = inf{cJ τA,n (σ(τA,n )), 1 − cτA,n (σ(τA,n )) :

q = 1, . . . , Q, τA,n ≤ M, n ∈ N, A ∈ Ak , A ∩ B2k (v) 6= ∅}.

(28)

12


The quantity ηM is a random variable due to the measurability of the temperature profile. In fact, T measurable implies that T (τA,n ) is a random variable, for A ∈ Ak and n ∈ N. The random variable ηM is positive Pk,α,β;T -a.s. by construction and for the positivity of T . We also define the finite (a.s.) random variable ¯ [ ¯ ¯ ¯ NM = ¯ Qx ∩ [0, M ].¯ (29) x∈B2k (v)

The joint probability law of the vector (ηM , NM ) conditioned to the event {TB4k (v) < M } is denoted by PηM ,NM . Again, ηM is positive and NM is finite PηM ,NM -a.s.. Then Z Pk,α,β;T (FM |TB4k (v) < M ) ≥

(ηM )NM dPηM ,NM > 0.

(30)

Therefore, by (27) and (30), we have Pk,α,β;T (FM ∩ {TB4k (v) < M }) > 0, where TB4k (v) depends on the sequence (UA,n : n ∈ N, A ∈ Ak ) while FM has positive J ,(A)

q conditional probability to be less or greater than cτA,n , for the independence of Jq ,(A) the U ’s and being cτA,n ∈ (0, 1). Among all the different systems we select one of them in the following way: A) All the interactions between two spins inside B2k (v) are equal to +1 with the exception of the interactions Jv,vi , where i = 1, . . . , dv /2 , that are equal to −1 (we are calling {v1 , . . . , vdv } the vertices adjacent to v); B) If e = {x, y} ∈ E with x ∈ B4k (v) \ B2k (v) and y ∈ B2k (v), the interaction Je is equal to σx (M ). We call such a system as the capital system, and denote as Σ(t) = (Σv (t) : v ∈ V ) ∈ {−1, +1}V the configuration of the capital system at time t. We notice that the capital system can coincide, in principle, with the original one. On it we consider a new set of Poisson processes (Pˆ{x} : x ∈ B4k (v)) with rate 1 that are active starting from time M . Therefore, all the arrivals of these Poisson processes are in [M, +∞). Conversely, if A 6= {x}, with x ∈ B4k (v), the Poisson processes PA ’s associated to the capital system remain the same. Moreover, (Pˆ{x} : x ∈ B4k (v)) are independent from all the other Poisson processes and also from the U ’s; we call Tˆx = (ˆ τ{x},n : n ∈ N) the set of arrivals of the Poisson process ˆ P{x} . We define the following stopping time related to the capital system:

ψˆk,v = sup{ˆ τ{x},1 ≥ M : x ∈ B4k (v)}, ˆ have an arrival (after i.e. ψˆk,v is the first time that all the Poisson processes P’s M ). Moreover, we also define a stopping time associated to the original system as: ψk,v = inf{τA,n ≥ M : A ∈ Ak , A ∩ B4k (v) 6= ∅, n ∈ N}, i.e. the quantity ψk,v identifies the first time of arrivals of the Poisson processes PA , with A ∩ B4k (v) 6= ∅, related to the original system, after time M . By independence of the P{x} ’s from PA ’s follows that, with positive probability, ˆ ψk,v ∈ [M, ψk,v ). Thus, for the capital system, there is a positive probability that in the interval of time [ψˆk,v , ψk,v ) (conditioned to ψˆk,v ∈ [M, ψk,v )) all the spins of the capital system inside B2k (v) have value constantly equal to +1. In fact, the temperature profile is positive, and all the spin flips have positive probability (analogously to (28)-(30) ). We call this event HM , in formula: n o n o HM = ψˆk,v ∈ [M, ψk,v ) ∩ Σu (t) = +1 : t ∈ (ψˆk,v , ψk,v ), u ∈ B2k (v) .


13

Therefore, by the strong Markov property and the independence of Pˆ{x} ’s from the others Poisson processes, we have Pk,α,β;T (IM ) > 0, where IM = {TB4k (v) < M } ∩ F M ∩ HM . + We now show that Σu (t) can remain equal to Σu (ψˆk,v ) with positive probability, for each t > ψˆk,v , for any u ∈ B4k (v) \ B2k (v), under the assumption that the event IM is satisfied. After ψˆk,v we change the coupling taking again the same set of Poisson processes for the original and the capital system, and different uniform random variables eA,n : A ∈ Ak , n ∈ N) for the capital system. Specifically, we set (U eA,n : A ⊂ B2k (v) 6= ∅, n ∈ N) are uniform random variables independent (i) (U from all the other random variables; eA,n = U (1) otherwise. (ii) U A,n We will see that, with positive probability, the capital system will remain equal to the original one for any t > ψk,v . (1) If A ∈ Ak is different from {v} and A ⊂ B2k (v) then, from the fact that the d-Egraph G is stable with respect to the vertex v (see Definition 1), one obtains that all the increments of the Hamiltonian at A are strictly positive, i.e. all the flips in A are in opposition of the Hamiltonian. By applying Corollary 3 and invoking the strong Markov property, we deduce that with positive probability Pk,α,β;T the capital system has no flips on A, for any A ⊂ B2k (v) and A 6= {v}. We are also using that these flips are e ’s in (i)). independent from the original system (see the definition of the U (2) Assume that (1) is true. If A ∈ Ak with A ⊂ B4k (v) and A ∩ (B4k (v) \ B2k (v)) 6= ∅ then, from the fact that in the capital system we have selected the interactions following point B) above, we obtain that that the increment of the Hamiltonian at A for the original system is smaller or equal to the one referred to the capital system. Therefore, the rate of a flip in A for (A) (A) J ,(A) the capital system, named cˆt , is such that cˆt ≤ ct . Thus, using also in the capital system the same UA ’s (for such A, see (ii)), they remain constant if the previous item is realized. c (3) Assume that (1) is true. If A ∈ Ak with A ∩ B4k (v) 6= ∅ and the first item is satisfied then the capital system and the original one have the same flips. It is worth noting that (2) and (3) are consequences of (1) that holds true with positive probability. We have proved that previous items are satisfied with positive probability. In the case of occurrence of such events one has that the selected vertex v of the capital system has, for any time t > ψk,v , all the neighbor spins equal to +1. Therefore, starting from the stopping time ψk,v , all the arrivals of P{v} have probability 1/2 to have a flip (independently one each other), so v, in the capital system, will change its value infinitely many times by Borel-Cantelli lemma. The last assertion is clearly a contradiction with the fact that ρI = 0, because in the capital system there is, by ergodicity, ρI > 0. Moreover re-sampling the interactions of the original system only over the finite set {e = {u, w} ∈ E : {u, w} ∩ B2k 6= ∅} there is a positive probability that it coincides with the capital system. Therefore, also in the original one, the quantity ρI must be larger than zero. This concludes the proof. ¤ Remark 3. The proof of the previous theorem depends on the the fact that the temperature is at any time different from zero.

14


We also notice that, the previous theorem can be generalized by considering the parameter α ∈ (0, 1) but fixing the initial configuration σ ∈ {−1, +1}V . In this case we can only prove, following the same line of the previous proof, that there exists a positive frequency of vertices that flip infinitely often. Hence, in general, we only know that |{x ∈ Bk (O) : x flips finitely many times}| lim inf > 0, k→∞ |Bk (O)| but the limsup could not coincide with the liminf and they could depend on the initial configuration σ. We can conclude that for any initial configuration σ the model is of type M or I. We now present a theorem in which we explore the possibility that ρI is larger than zero without requiring the positivity of the temperature profile. The basic assumption is ρF > 0. We address the reader to the next section for sufficient conditions to obtain that some specific sites fixate. Theorem 5. Let us consider k ∈ N, β ∈ [0, 1], a d-graph G = (V, E) with d ∈ N and T a temperature profile fast decreasing to zero. For a given α ∈ (0, 1) suppose that there exists a finite set R ⊂ V , and a finite set S ⊂ E such that there exists JS ∈ {−1, +1}S with the following properties: (i) For all J ∈ {−1, +1}E coinciding with JS on S, Pk,hJ i,β;T ({there exists lim σu (t), ∀ u ∈ R}) > 0. t→∞

ΣJ R

R

We denote by ⊂ {−1, +1} {−1, +1}R such that

the set collecting σ ˆR = (ˆ σu : u ∈ R) ∈

Pk,hJ i,β;T ({ lim σu (t) = σ ˆu , ∀ u ∈ R}) > 0. t→∞

ΣJ R

(ii) There exists σ ˆR ∈ and a finite set R0 ⊂ V such that: for any given σ coinciding with σ ˆR over R, there exists D ∈ Ak , D ⊂ R0 with ∆D HJ (σ) ≤ 0. Then ρI > 0, and hence the model is of type M (being ρF > 0 by hypothesis). Proof. First of all, we notice that there exists J ∈ {−1, +1}E coinciding with JS on S with positive probability, being S finite and α ∈ (0, 1). By item (i) we have that ΣJ R 6= ∅. Item (ii) allows us to select σ ˆR such that there exists a finite set of D’s flipping with probability not smaller than 1/2 at the arrival of the Poisson PD . In fact, R0 is finite and by using the pigeonhole principle, there exists D ∈ Ak , with D ⊆ R0 , such that infinitely many times ∆D HJ (σ) ≤ 0. Therefore, for the Levy’s form of Borel-Cantelli Theorem, there are infinite flipping in D Pk,α,β;T -a.s.. By ergodicity, we have ρI > 0. ¤ Remark 4. We notice that previous result seems to have a purely theoretical flavor. However, we have some examples in which one can apply Theorem 5. We believe that the most relevant ones are those where R is a separating set so that the internal part of R is only influenced by the value of the field on the sites of R. By hypothesis the sites on R will fixate with positive probability and, under this condition, the field defined on the internal and the external sites of R becomes conditionally independent (see Figure 1 for an example). 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, β; T )-model. It is known that the (1, α, β; T )-model on the hexagonal lattice is of type F when T ≡ 0, see [24] and in the next section this result is proven when T is fast decreasing (see item a. of


15

Figure 1. It is here represented a region of the periodic graph G embedded in R2 . For the notation, see Theorem 5. Consider k = 1, α ∈ (0, 1), β ∈ (0, 1] (resp. β ∈ [0, 1)) and T fast decreasing to zero. Set the black bullets as sites with spin +1 (resp. −1) and the other spins can be arbitrarily selected. The set R is formed by the black bullets. The set S is given by all the edges connecting a black bullet with another black bullet or with the white one. The continuous line represents an edge e with Je = +1, while the dotted line is associated to Je = −1. The spins over R cannot flip, and remain constant at +1 (resp. −1) if initially taken positive (resp. negative). D is formed by the cental white bullet, and at each arrival of the Poisson PD it has probability 1/2 to flip. Hence, ρI > 0. Notice that the only interactions which really matter are those in S, while the other ones can be arbitrarily selected.

Theorem 7). It is important to notice that the hexagonal lattice in our framework is a 2-graph. In the following result we show that a (k, 1, 1/2; T )-model with k ≥ 2 cannot belong to F. The question if it is of kind M or I remains open. Theorem 6. The (k, 1, 1/2; T )-model on the hexagonal 2-graph GH = (VH , EH ) with k ≥ 2 is not of type F. Proof. By contradiction suppose that the model is of type F. Therefore there exists, for any x ∈ VH , σx (∞) = lim σx (t) Pk,1,1/2;T − a.s. t→∞

We denote with σ(∞) = (σx (∞) : x ∈ VH ) the random limit configuration. By ergodicity and being β = 1/2 (hence the symmetry, at any time, between the signs of the spins) we obtain that 1 |{x ∈ Bk (v) : σx (∞) = +1}| = Pk,1,1/2;T − a.s. (31) k→∞ |Bk (v)| 2 for any v ∈ VH . As in [24] we define the domain wall DH that is a subset of the hexagonal dual lattice, which separates the positive and negative spins joining the center of the hexagons (see Figure 2). By (31) DH 6= ∅ Pk,1,1/2;T -a.s.. The dual lattice is formed lim

16


ascendent y1

y x1

vertical

z

y2 x

x

y

x2 descendent

Figure 2. The hexagonal lattice with the domain wall DH in bold. The squares and the bullets represents spins with opposite signs in the two situations of 2π/3 (left panel) and π/3 (right panel) angles.

by three classes of edges: the vertical edges, the ascending ones and the descending ones (see Figure 2). As in (31), there exists the density ρver (ρasc , ρdes , resp.) of the vertical (ascending, descending, resp.) edges of the domain wall (analogously to (31)). By symmetry and being DH 6= ∅, we have that ρver = ρasc = ρdes > 0 Pk,1,1/2;T -a.s.. On each site v ∈ VH with σv = −1 we construct a closed triangle connecting the centers of the three hexagons containing v. We denote S as Φv,H such triangle, and introduce the set containing all such triangles ΦH = v∈VH :σv =−1 Φv,H . The set DH can be viewed as the boundary of the set ΦH . Simplicial homology arguments (see [27]) gives that the domain wall DH can be decomposed in connected components, each of them belonging to one of the following families: • Closed component: the component is given by a sequence of edges forming a cycle. • Bidirectional infinite component: each edge belonging to the component has two adjacent edges. Two adjacent edges of the set DH may form an angle of π, 2π/3 and π/3. If there exists a closed component, then there exists at least one couple of adjacent vertices forming an angle different from π. In this case, by ergodicity, the sum of the densities of angles 2π/3 and π/3 is different from zero. If all the components are of type bidirectional infinite, then we have two subcases. One of the components has an angle different from π or, otherwise, the component is a straight line (for example, with all vertical edges). In the former case, we invoke ergodicity and obtain that, as in the closed case, the sum of the densities of angles 2π/3 and π/3 is different from zero. In the latter case, ergodicity and symmetry guarantee the existence of another component of DH which is a straight line as well (with not vertical edges). The intersection of the two components forms an angle different from π, and we fall in the previous case. 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} ∈ EH such that the random time T{x,y,z} = inf{t ∈ R+ : ∀t0 ≥ t, σx (t0 ) = σz (t0 ) = −σy (t0 )} is finite Pk,1,1/2;T -a.s.. However, there are infinite arrivals of the Poisson P{y} in the random interval (T{x,y,z} , +∞) and at each arrival the spin y flips with a probability at least 1/2 (the value of such probability depends on the temperature at the arrival time, and it is one in the case of zero-temperature). By Levy’s conditional form of


17

the Borel-Cantelli lemma (see [30]), we have a contradiction and the (k, 1, 1/2; T )model is of type M or I. If the density of the angles π/3 is zero (see Figure 2, left panel), then we have only angles of 2π/3 Pk,1,1/2;T -a.s.. In this case there exists an edge {x, y} ∈ EH and its four adjacent edges {x1 , x}, {x2 , x}, {y1 , y}, {y2 , y} ∈ EH such that the random time T{x,y,x1 ,x2 ,y1 ,y2 } = 0

0

= inf{t ∈ R+ : ∀t ≥ t, σx1 (t ) = σy1 (t0 ) = −σx2 (t0 ) = −σy2 (t0 ) and σx (t0 ) = σy (t0 )} is finite Pk,1,1/2;T -a.s.. Analogously to the previous case, there are infinite arrivals of the Poisson P{x,y} in (T{x,y,x1 ,x2 ,y1 ,y2 } , +∞) and at each arrival the set {x, y} flips with a probability equals to 1/2. Levy’s conditional form of the Borel-Cantelli lemma leads to a contradiction and the (k, 1, 1/2; T )-model cannot be of type F. ¤ Theorem 6 and other examples not reported here suggest that increasing the value of k does not facilitate the fixation of the sites. Furthermore, we do not have in mind examples going in the opposite direction, i.e. the growth of the value of k let a model becomes, for example, of type I from an original type F. In this respect, see the conjecture in the concluding section. 4. Conditions for ρF > 0 In this section we give some sufficient conditions to obtain ρF > 0. Initially we present a new definition that will be useful in the sequel. For a given d-graph G = (V, E) and a finite set Λ ⊂ V , we define the quantity hG,Λ =

|{{x, y} ∈ E : x, y ∈ Λ}| . |Λ|

(32)

For a given finite set Λ ⊂ V , for J ∈ {−1, +1}E , σ ∈ {−1, +1}V , we also set X HΛ,J (σ) = − Je σx σy (33) e={x,y}∈E:x,y∈Λ

and the density energy on Λ as: DΛ,J (σ) =

HΛ,J (σ) . |Λ|

(34)

The following inequalities will be useful and they can be checked by the reader lim inf lim inf t→+∞ k→∞

HBk (O),J (σ(t)) ≥ DJ = |Bk (O)|

= lim inf min DBk (O),J (σ) ≥ − lim inf hG,Bk (O) ≥ −dG /2 > −∞. k→∞

σ

k→∞

H

(σ(t))

(35)

(O),J Formula (35) guarantees that Bk|B > −∞. In particular, if T is fast k (O)| decreasing to zero, it does not exist a site whose flips are in favor of the hamiltonian infinitely many times. We notice that only in the case of a temperature profile T that does not satisfy Corollary 2 one can hope to obtain ρF > 0. Hence a very simple necessary condition for ρF > 0 is lim inf t→∞ T (t) ln t ≤ 4dG . In the following we work under the stronger assumption that the temperature profile T is fast decreasing to zero and we give, jointly to it, some sufficient conditions to obtain ρF > 0. Next theorem provides two conditions with very simple hypotheses, where it is presented an extension to the case k > 1 of previous results on k = 1 (see e.g. [14, 24]).

18


Theorem 7. Let us consider a d-graph G = (V, E) with d ≥ 1 and a (k, α, β; T )model with k ∈ N, α, β ∈ [0, 1] and a temperature profile T fast decreasing to zero. If at least one of the following conditions is satisfied: a. There exists v ∈ V such that for any A ∈ Ak with v ∈ A the cardinality of the set ΓA = {{u, w} ∈ E : u ∈ A, w ∈ Ac } is odd. b. The parameters α, β belong to (0, 1]. There exists a finite connected set A ⊂ V such that |A| > k and for any B ∈ Ak with B ∩ A 6= ∅ one has |{{u, v} ∈ E : u ∈ B ∩ A, v ∈ B c ∩ A}| > |{{u, v} ∈ E : u ∈ B, v ∈ B c ∩ Ac }| + |{{u, v} ∈ E : u ∈ B ∩ Ac , v ∈ B c ∩ A}|. (36) Then ρF is larger than zero. Specifically: a. limt→∞ σv (t) exists Pk,α,β;T -a.s.; b. Pk,α,β;T ({limt→∞ σu (t) there exists , ∀ u ∈ A}) > 0. Proof. We prove the theorem by separating the cases. a. As in [24, 14] we see that every time that there is a flip in A ∈ Ak with v ∈ A, then such a flip is in favor or in opposition of the hamiltonian. In fact, the number of edges in the set {{u, v} ∈ E : u ∈ A, v ∈ Ac } is odd and for any choice of the interactions J ∈ {−1, +1}E it is not possible to obtain ∆A HJ (σ) = 0. Only a finite number of flips in opposition of the hamiltonian are allowed Pk,α,β;T -a.s. because the temperature profile is fast decreasing to zero; moreover, for the vertex v, also the number of flips in favor of the hamiltonian must be finite Pk,α,β;T -almost surely. Indeed, if it is not the case, we should obtain by ergodicity, that the quantity DJ , defined in (35), should be equal to −∞. This contradicts what pointed out in (35). b. First, we define the set E(C) ⊂ E as E(C) = {{u, v} ∈ E : u, v ∈ C}, for any C ⊂ V . Now consider A as in the statement in item b. of the theorem. Since α > 0 and E(A) is finite, with positive probability Je = +1 for any e ∈ E(A). In T fact the independence leads to Pk,α,β;T ( e∈E(A) {Je = +1}) = α|E(A)| > 0. T Analogously, being β > 0, it is Pk,α,β;T ( v∈A {σv (0) = +1}) = β |A| . By inequality (36), we have that ∆A HJ (σ) ≥ 1, when σ coincides with σ(0) over the entire set of vertices of A, i.e. when σv = +1 for v ∈ A. Let us take a B as in the statement of the theorem. Hence, by Corollary 3, we do not have flips at any time on B with positive probability. Therefore, there is a positive probability that in the region A all the spins maintain the initial value +1. By ergodicity, we obtain ρF > 0. ¤ Remark 5. In Theorem 7 item a., we obtain the condition dx odd in the case of k = 1 for having ρF > 0 (see [24]). Furthermore, for k = 1, (36) becomes |{{u, v} ∈ E : u ∈ B ∩ A, v ∈ B c ∩ A}| > |{{u, v} ∈ E : u ∈ B, v ∈ B c ∩ Ac }|. Example 1. For k = 1 we can take the hexagonal planar lattice as in [17]. However, for k = 2, it is easy to check that item a. of Theorem 7 is not verified for such graph. It is not difficult to provide examples for k > 1 for obtaining ρF > 0, and we present here a graph as an example on which item a. of Theorem 7 can be applied when k = 2 (see Figure 3). For what concerns item b. of Theorem 7, we address the reader to Figure 1. In


19

v

Figure 3. All the intersections among lines represent sites of the graph. The site denoted with the black bullet is v. Each set A 3 v of cardinality 2 is such that |ΓA | = 5. The case A = {v} is such that |ΓA | = 3.

this case we refer to k = 1. Consider A the set of the black bullets, and B the set containing only a vertex of A. It is clear that item b. of Theorem 7 holds. In the following results we adapt the definition of e-absent set and the strategy of the proof of ρF > 0 in [2, 14]. Definition 3. For a (k, hJ i, β; T )-model on a d-graph G = (V, E) and for a finite subset C of V we say that σC = (σx : x ∈ C) is k -absent if for any σC c = (σx : x ∈ C c ) there exists a finite sequence of m flips, whose j-th flip occurs in A(j) ⊂ C, A(j) ∈ Ak , such that ∆A(m) HJ (σ (m) ) < ∆A(j) HJ (σ (j−1) ) = 0, for each j = 1, . . . , m, where σ (0) coincides with σC in C and σC c in C c . Moreover, we set (j) σ (j) = (σ (j−1) )(A ) . Remark 6. For any given J , if σC is k-absent, then it is also k 0 -absent, for each k 0 > k, being Ak ⊂ Ak0 . We notice also that, if σC is k-absent and if σ ˆC is such that, for any σC c , there exists a finite sequence of m ˆ flips A(j) ⊂ C, A(j) ∈ Ak , with ∆A(j) HJ (ˆ σ (j) ) ≤ 0, (m) ˆ for each j = 1, . . . , m, ˆ and σ ˆC = σC , then σ ˆC is k-absent as well. Moreover, we notice that σC is k-absent only on the basis of the restriction of J to the edges {{x, y} ∈ E : {x, y} ∩ C 6= ∅}. Theorem 8. For a (k, hJ i, hσi; T )-model with fast decreasing temperature profile T on a d-graph G = (V, E), any k-absent configuration takes place only on a finite time interval Pk,hJ i,hσi;T -almost surely. Proof. Formula (35) guarantees that lim inf lim inf t→+∞ k→∞

HBk (O),J (σ(t)) > −∞. |Bk (O)|

Hence, by ergodicity, it is not possible to find a site whose flips are infinitely often in favor of the hamiltonian. ¤ Theorem 9. For a (k, hJ i, hσi; T )-model with fast decreasing temperature profile T on a d-graph G = (V, E), if there exists C ⊂ V and u, v ∈ C with νG (u, v) ≥ k such that all the configurations σC with σu = σv (resp. σu = −σv ) are k-absent, then limt→∞ σu (t) = − limt→∞ σv (t) (resp. limt→∞ σu (t) = limt→∞ σv (t)) Pk,hJ i,hσi;T almost surely.

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-3

-1

-4

2

0

3

9

11

6

1

-2

7

4

5

8 10

C

Figure 4. These sites constitute a region of a graph G = (V, E) with V = L1 . The continuous line stands for an edge e with Je = +1, while the dotted line describes Je = −1. The black bullets and squares are spins fixing definitively at a constant configuration (+1 or −1) and, in general, bullets can be different from squares. When bullets and squares share the same constant configuration, then sites 2 and 4 flip infinitely often, i.e. it does not exist limt→∞ σj (t), with j = 2, 4. Differently, when bullets and squares have opposite signs, then it does not exist limt→∞ σj (t), with j = 3, 5. All the configurations σC having σ6 = −σ11 are 1-absent.

Proof. We only prove the case of σu = σv , being the proof of the case σu = −σv analogous. The vertices u and v, defined in the theorem, cannot flip simultaneously almost surely because νG (u, v) ≥ k. In fact, for any A ∈ Ak , the set {u, v} is not a subset of A. By Theorem 8, we know that all the configurations having σu (t) = σv (t) disappear. Therefore there exist limt→∞ σu (t), limt→∞ σv (t) and they have opposite sign. ¤ Corollary 10. For a (k, α, β; T )-model on a d-graph G = (V, E), with α ∈ (0, 1) and temperature profile T fast decreasing to zero, if there exists a finite C ⊂ V and u, v ∈ C with νG (u, v) ≥ k such that all the configurations σC with σu = σv (resp. σu = −σv ) are k-absent for a choice of J on the edges {{x, y} ∈ E : {x, y}∩C 6= ∅}, then ρF > 0. Proof. The choice of J on {{x, y} ∈ E : {x, y} ∩ C 6= ∅}, that let the configurations σC with σu = σv be k-absent, is realized with positive probability from the fact that the set {{x, y} ∈ E : {x, y} ∩ C 6= ∅} is finite. Therefore, there is a positive probability that we are in the conditions of Theorem 9. By ergodicity we obtain that the limit, defining ρF , exists and it is positive. ¤ We notice that in the case of a (k, α, hσi; T )-model the quantity, similar to ρF , |{x ∈ Bk (v) : x flips finitely many times}| k→∞ |Bk (v)| is larger than zero but we cannot use the ergodicity and so we can not deduce that there exists ρF . We present an example in which Theorem 9 and Corollary 10 can be applied. lim inf

Example 2. We consider k ≤ 3 and a 1-graph G = (V, E), as in Figure 4, with 1 vertices (the cell will be scaled by 15 and translated by − 10 ) given by VCell = {1, 2, 3, 4, 5} and edges ECell = {{−1, 1}, {0, 1}, {1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}}. We consider as C of Definition 3 the vertices {1, 2, . . . , 11}. The interactions between the elements of C are taken all positive with the exception of J{2,4} = −1. By contradiction assume that all the spins of C flip infinitely often. In particular,


21

Figure 5. Representation of Γ3,3 (L2 ).

it should exists a random sequence (Sn : n ∈ N) with limn→∞ Sn = ∞ such that σ6 (Sn ) = −σ11 (Sn ) for each n and νG (6, 11) = 3. We now show that all the configurations σC having σ6 = −σ11 are 1-absent (by Remark 6 also k-absent for any k). Thus by Theorem 9 and Corollary 10 we have a contradiction obtaining ρF > 0. By symmetry of the dynamics with respect to global flips, i.e. σ → −σ, we can take σ6 = +1 and σ11 = −1. In principle one should check 29 configurations σC to be 1-absent. However we notice that there exists a sequence of single flips in favor or indifferent for the hamiltonian leading to σ7 = σ8 = −1. This fact will be useful in showing the 1-absent property for configurations σC . We now distinguish the cases of σ1 = +1 and σ1 = −1. Consider σ1 = +1. (1) If σ4 = −1 then, by flipping the site 6, we obtain ∆{6} HJ (σ) ≤ −4 (where σ coincides with σC on C); (2) if σ4 = +1 and σ2 = +1 then, by flipping the site 4, we have ∆{4} HJ (σ) = 0 and we are in the case described by the previous item; (3) if σ4 = +1 and σ2 = −1 then, by flipping the site 2, we have ∆{2} HJ (σ) = 0 and we are in case (2). The case σ1 = −1 is analogous to the previous one by replacing site 2 with 3 and site 4 with 5. Therefore limt→∞ σ6 (t) = limt→∞ σ11 (t) and this also implies that limt→∞ σi (t) = limt→∞ σ6 (t), for i = 7, 8, 9, 10. We notice that with positive probability all the interactions presented in the figure are positive with the exception of J{2,4} = −1. In this case we can apply the previous argument to the first and third part of the graph (see Figure 4). We have that limt→∞ σi (t) = limt→∞ σ1 (t), for i = −4, −3, −2, −1, 0. If limt→∞ σ1 (t) = limt→∞ σ6 (t) then the sites 2 and 4 flip infinitely many times. Differently if limt→∞ σ1 (t) = − limt→∞ σ6 (t) then the sites 3 and 5 flip infinitely many times. This framework can be though as an illustrative example also of Theorem 5. In fact, the set R can be taken as coinciding with {−4, −3, −2, −1, 0, 1, 6, 7, 8, 9, 10, 11}, S is the set of the edges represented in Figure 4, ΣJ R contains at least one among the four combinations of black bullets ±1 and black squares ±1, and accordingly D is given by {2, 4} (bullets and squares with the same sign) or {3, 5} (otherwise). 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). In particular, for any k, d ∈ N we can provide examples in which we use Theorem 9 and Corollary 10 to obtain models with ρI and ρF larger than zero.

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Remark 7. In [14] the Ising stochastic model at zero temperature on Z2 is studied. Their model can be rewritten in our language as (1, α, β; T )-model on the 2-graph L2 = (Z2 , E2 ) with α ∈ (0, 1), β = 1/2 and T ≡ 0. In [14] it is shown, by an elegant argument, that the model is of type M. In [24] for the same model with deterministic interactions J ≡ +1, i.e. α = 1, it is shown that the model is of type I. With a simple coupling argument one can prove that also J ≡ −1, i.e. α = 0, the system is of type I. In fact consider two systems on the d-graph Ld = (Zd , Ed ) with J ≡ +1 and J ≡ −1 and β = 1/2. We say that a vertex v = (v1 , . . . , vd ) ∈ Zd Pd is even (resp. odd) if | i=1 vi | is even (resp. odd). We take a initial configuration σ for the system with J ≡ +1 and, for the system with J ≡ −1, we take the initial configuration σ ˜ ½ σv if v is even; (37) σ ˜v = −σv if v is odd. Therefore also the random variables (˜ σv ; v ∈ Zd ) is a collection of i.i.d. Bernoulli random variables with β = 1/2. Taking the same Poisson processes and the same random variables U ’s for both the systems at any time t one obtain ½ σv (t) if v is even; σ ˜v (t) = (38) −σv (t) if v is odd. Therefore the two systems are of the same type (I, F or M). The same construction works for any bipartite graph. Example 2, with specific reference to Figure 5, suggests a general result on a particular class of graphs. We firstly define such class, and then present a simple and meaningful theorem. Definition 4. Fix ` ≥ 2 and m ≥ 1, and consider a d-graph G = (V, E), with d ≥ 1. We denote as Γ`,m (G) the graph such that each edge of G is replaced with m identical cycles whose number of edges is 2` with the following property: (i) two adjacent cycles have only one vertex in common (we denote such vertices as common vertices); (ii) the distance between two common vertices u and v ∈ Γ`,m (G) νΓ`,m (G) (u, v) is equal to `. In Figure 5 we have the representation of Γ3,3 (L2 ). It is easy to check that Γ`,m (G) is a d-graph, for ` ≥ 2 and m ≥ 1. Theorem 11. Let G = (V, E) be a d-graph, with d ≥ 1. Consider a (k, α, β; T )model on Γ`,m (G) with α ∈ (0, 1) and fast decreasing temperature profile T . If ` ≥ k ∧ 2 and m ≥ 3, the model is of type M. We omit the proof, and address the reader to Example 2. However, we provide a sketch of the employed strategy. The statement of Theorem 11 allows to find three adjacent cycles (two external cycles and the middle one) such that the interactions are all +1 with the exception of only one of them in the middle cycle. Then, one can prove that the vertices in the external cycles fixate, hence leading to the presence of a site flipping infinitely many times in the middle cycle. Therefore, there is a fraction in (0, 1) of fixating sites and the remaining part of sites flipping infinitely many times. This means that the model is of type M. When G = Ld , further arguments can be provided. Theorem 12. Let G = Ld , for d ≥ 2. Consider a (k, α, β; T )-model on Γ`,m (G) and fast decreasing temperature profile T . If d = 2, α ∈ {0, 1} and β = 1/2, then the (k, α, β; T )-model is of type I.


23

The proof is omitted, in that it derives by adapting the proof of Theorem 2 in [24] to the graphs Γ`,m (L2 ). The strategy is to use the original vertices of the graph (the vertices x’s such that dx = 8) and proceeding by contradiction. Theorem 13. Let G = Ld , for d ≥ 2. Consider a (k, α, β; T )-model on Γ`,m (G). If α = 1, then there exists ² ∈ (0, 1/2) and a temperature profile T fast decreasing such that: for each β ∈ [0, ²) ∪ (², 1], then the (k, α, β; T )-model is of type F. Also in this case we omit the proof, which is of remarkable length. Indeed, it can be obtained by mimicking [13] and estimating in our specific graph the upper bounds that a given site is −1 for a large value of t. 5. 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 very general periodic graphs embedded in Rd . The obtained results offer a picture of the behavior of the stochastic Ising models, and they can be compared with the standard case of zero temperature, cubic lattice Ld and k = 1. For the cubic lattice L2 the paper [24] says that for α = 1 or α = 0 and β = 21 the model is of type I. On the same graph, [14] prove that the model is of type M when α ∈ (0, 1) and β = 21 (actually their arguments hold true for β ∈ (0, 1)). In [13, 23] it is shown that, for Ld with d ≥ 2, the model is of type F for α = 1 and β sufficiently close to one (or, by symmetry, sufficiently close to zero). In particular it is known that the limit configuration is done of all +1 (or, by symmetry, −1). For a better visualization of the results, see Figure 6 (left panel). For what concerns the graphs Γ`,m (G) of Definition 4, see Figure 6 (central panel). In this case, when ` ≥ k ∧ 2 and m ≥ 3, we have shown that the (k, α, β; T )model is of type M for α ∈ (0, 1) and β ∈ [0, 1]. When G = Ld , α = 1 and β is sufficiently close to one (or, by symmetry, sufficiently close to zero), then the (k, α, β; T )-model is of type F. The particular case of Γ`,m (L2 ) gives that the the (k, α, β; T )-model is of type I for α = 0, 1 and β = 1/2. When T is fast decreasing to zero and positive, and by considering graphs Ld with d ≥ 2, then it is possible to exclude that the (k, α, β; T )-model is of type F, refer to Figure 6 (right panel). We feel that Figure 6 might contribute to highlight some problems left open by this paper. To conclude, we also think that our results may represent a first move towards the following two conjectures. (i) If α, β ∈ (0, 1) and T is fast decreasing to zero, then the type of the (k, α, β; T )-model over a d-graph G = (V, E) is identified by the value of k ≥ 1 and by the graph G. (ii) Fix α, β ∈ [0, 1], a temperature profile fast decreasing to zero T and a d-graph G. Set F, M and I to 1,2,3, respectively. Define the function ξα,β,T,G : N → {1, 2, 3} assigning to a value of k ≥ 1 the type of the (k, α, β; T )-model over G. The function ξα,β,T,G is nondecreasing. References [1] R. Arratia. Site recurrence for annihilating random walks on Zd . Ann. Probab., 11(3):706–713, 1983.

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M F

I

(d=2) M

I

M or I F

I (d=2)

M

F

M or I

I(d=2)

F

M

M or I

Figure 6. Graphical visualization of the conclusions. The panel on the left describes the main results in the literature for the stochastic Ising models at zero-temperature on the graph Ld . Notice that the case of β = 1/2 holds only for d = 2. The central panel represents our results for the graphs Γ`,m (Ld ). The panel on the right depicts Theorem 4.

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