Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees
EErgodicity versus non-ergodicity for Probabilistic CellularAutomata on rooted trees
Bruno Kimura
Delft Institute of Applied Mathematics [email protected]
Technische Universiteit Delft, Van Mourilk Broekmanweg 6, 2628 XE Delft , The Netherlands
Wioletta Ruszel
Delft Institute of Applied Mathematics
Technische Universiteit Delft, Van Mourilk Broekmanweg 6, 2628 XE Delft , The Netherlands
Cristian Spitoni
Institute of Mathematics
Universiteit Utrecht, Budapestlaan 6, 3584 CD Utrecht , The Netherlands
February 1, 2019
Abstract
In this article we study a class of shift-invariant and positive rate probabilistic cellular automata(PCAs) on rooted d-regular trees T d . In a first result we extend the results of [13] on trees, namely weprove that to every stationary measure ν of the PCA we can associate a space-time Gibbs measure µ ν on Z × T d . Under certain assumptions on the dynamics the converse is also true.A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCAon d-ary trees for d ∈ { , , } and characterizing the invariant Bernoulli product measures. Cellular Automata (CAs) are discrete-time dynamical systems on a spatially extended discrete space.They are well known for being easy to implement and for exhibiting a rich and complex nonlinear behaviouras emphasized for instance in [39, 41, 17, 16]; furthermore, they can give rise to multiple levels of orga-nization [15]. Probabilistic Cellular Automata (PCAs) whose the updating rule is now considered to bestochastic, see [34], are a straightforward generalization of CAs and are employed as modeling tools in awide range of applications, e.g. HIV infection [29], biological immune system [36], weather forecast [9], heartpacemaker tissue [26], and opinion forming [1]. Moreover, a natural context in which the PCAs main ideasare of interest is that of evolutionary games [30, 31, 32].Strong relations exist as well between PCAs and the general equilibrium statistical mechanics frame-work [40, 28, 18, 7, 12, 13, 14, 33, 37, 38]. A central question is the characterization of the equilibriumbehavior of a general PCA dynamics. For instance, one primary interest is the study of its ergodic prop-erties, e.g. the long-term behavior of the PCA and its dependence on the initial probability distribution.Regarding the ergodicity for PCAs on infinite lattices, see for instance [37] for details and references. More-over, conditions for ergodicity for general PCAs can be found in the following papers: [18, 10, 20, 24, 25, 35].Furthermore, in case of a translation-invariant PCA on Z d with positive rates, it has been shown in [13]that the law of the trajectories, starting from any stationary distribution, is given by a Gibbs state for somespace-time associated potential (in Z d +1 ). Moreover, it has also been proven that the converse is true: allthe translation-invariant Gibbs states for such potential correspond to statistical space-time histories for thePCA. Therefore, phase transition for the space-time potential is closely related to the PCA ergodicity in thesense that non-uniqueness of translation invariant Gibbs states is equivalent to non-uniqueness of stationarymeasures for the PCA. The main ingredient for proving this result is the use of the local variational principlefor the entropy density of the Gibbs measure. However, as it has been proved in [2], the variational principle1 a r X i v : . [ n li n . C G ] J a n or Gibbs states fails for nearest neighbor finite state statistical mechanics systems on 3-ary trees. Hence,a first result to this paper is to extend the results presented by [13] for a class of PCAs on infinite rootedtrees. In particular, the PCAs considered in this paper have positive rate shift-invariant local transitionprobabilities such that each local probabilistic rule depends only on the spins of the children of the node.This class of PCAs has generally a Bernoulli product measure as an invariant measure, and they are thenatural generalization on trees of the models considered in [22].A second type of results in this paper is to give conditions for ergodicity in case of d -ary trees, with d ∈ { , , } . Our positive rate PCAs satisfy indeed such conditions (i.e. (3.1) and (3.3)) that, when iteratingthe dynamics from the Bernoulli product measure, the resulting space-time diagram defines non-trivialrandom fields with very weak dependences. This fact allows us to give a detailed analysis of the ergodicityproblem and, for two relevant examples of PCA dynamics, we are able to find the critical parameters.The paper is organized as follows. In Section 2 we extend the results of [13] in case of infinite rooted d -ary trees. We first define the PCA on a countably infinite set and in this general framework we show howstationary measures for a PCA can be naturally associated to Gibbs measures (Theorem 2.1). In order tostate the converse result, we first restrict ourselves to the case of infinite rooted trees and to PCAs withnondegenerate shift-invariant local transition probabilities that depends only on the spins of the childrenof the node. For this class of PCAs, we state that all the time-invariant Gibbs states for the potentialcorrespond to statistical space-time histories for the PCA (Theorem 2.2). In Section 3 we give resultsconcerning conditions for the ergodicity of the PCA on d -ary trees. First we characterize Bernoulli productstationary measures via Lemma 3.1. In Theorem 3.1 we show that for d = 1 the PCA is always ergodic,and the same occurs for d = 2 with the additional assumption of spin-flip symmetry of the local transitionprobabilities. In Theorem 3.2 the case of d = 3 is studied. We give two examples (Section 3.1.1, Section 3.1.2)where the critical parameters can be computed. Section 5 and the Appendices are devoted to the proofs ofthe main results. Let the single spin space be a nonempty finite set S and let V denote a countably infinite set (for example,the d -dimensional cubic lattice Z d or, more generally, the vertex set of a countably infinite graph). In thefollowing we introduce a special class of discrete-time Markov chains on the state space Ω = S V whosemain feature is the fact that given the previous configuration, for the next one all spins are simultaneouslyupdated accoding to independent local transition probabilities ( parallel updating ), the so-called probabilisticcellular automata.We define the probabilistic cellular automaton as follows. Definition 2.1.
A PCA is a discrete-time Markov chain on Ω with the following properties. At each site i in V (a) corresponding to each configuration x ∈ Ω we associate a probability measure p i ( ·| x ) on S , and(b) assume that for every spin s , the map x p i ( s | x ) is a local function. So, there is a finite subset U ( i ) of V such that the equality p i ( s | x ) = p i ( s | y ) holdsfor every s whenever x and y satisfy x j = y j for each j in U ( i ) .In this setting, we associate to each point x in Ω the product measure P ( dy | x ) = O i ∈ V p i ( dy i | x ) , (2.1) and introduce the probabilistic cellular automaton dynamics on our state space Ω by considering the Markovkernel P given by the expression P ( x, B ) = P ( B | x ) (2.2) where B is a Borel set of Ω . Now, we recall the definition of a stationary measure for the dynamics P . Definition 2.2.
A probability measure ν on Ω is called stationary for the dynamics P defined above if Z P ( x, B ) ν ( dx ) = ν ( B ) holds for every Borel set B of Ω . .2 From PCA to Gibbs measures... In this section we will show how stationary measures for a PCA can be naturally associated to Gibbsmeasures for a corresponding equilibrium statistical mechanical model. Let us consider the set of sites givenby the countably infinite set Z × V , the collection S consisting of all nonempty finite subsets of Z × V . Wealso consider the configuration space Ω = S Z × V together with its product σ -algebra F . Given an arbitraryspace-time spin configuration ω in Ω, for each site x in Z × V , say x = ( n, i ), let ω n,i denote the value ω x ofthe spin at this site, just for simplicity. Furthermore, for each integer n and each configuration ω , we definethe configuration at time n as the element ω n of Ω given by ω n = ( ω n,i ) i ∈ V .Now, let us consider again the setting from the previous section. We will assume that the PCA dynamicsis nondegenerate, that is, the local transition probabilities have positive rates: p i ( s | x ) > i ∈ V , s ∈ S and x ∈ Ω . Furthermore, we also suppose that for each site i , the set { j ∈ V : i ∈ U ( j ) } (2.3)is finite, which means that at each step in the dynamics of the PCA, each spin can have influence only onthe future state of a finite number of spins. Given a stationary measure ν for P , it is possible to constructa probability measure µ ν on (Ω , F ) uniquely determined by the identity µ ν ( ω t ∈ B , ω t +1 ∈ B , . . . , ω t + n ∈ B n ) = Z B ν ( dx ) Z B P ( x , dx ) · · · Z B n P ( x n − , dx n ) , (2.4)where t is an integer, n a positive integer, and B , B , . . . , B n are Borel sets of Ω . In the following, given asite x in Z × V , say x = ( n, i ), we will use U ( x ) to denote the set U ( x ) = { ( n − , j ) : j ∈ U ( i ) } . Observe that our assumption (2.3) is equivalent to say that for each point x , the set { y ∈ Z × V : x ∈ U ( y ) } is finite. This remark is very useful for proving the next theorem, whose proof is given in Appendix A. Theorem 2.1.
The space-time measure µ ν obtained from a stationary measure ν for the PCA is a Gibbsmeasure for the interaction Φ = (Φ A ) A ∈ S , where each Φ A : Ω → R is given by Φ A ( ω ) = ( − log p i ( ω x | ω n − ) if A = { x } ∪ U ( x ) for some x = ( n, i ) , otherwise. (2.5) We specify now the class of PCAs that will be considered in this paper. We introduce indeed probabilisticcellular automata on d -ary trees V = T d with root o and degree deg( x ) = d + 1 for all vertices x = o anddeg( o ) = d . Without loss of generality, the d -ary tree T d can be regarded as the set [ n ≥ { , . . . , d − } n consisting of all finite sequences of integers from 0 to d −
1. Given finite sequences i in { , . . . , d − } n and j in { , . . . , d − } m , say i = ( i k ) n − k =0 and j = ( j k ) m − k =0 , we naturally define their sum i + j as the concatenationof these sequences, i.e., the sum is the element of { , . . . , d − } m + n given by( i + j ) k = ( i k if k ∈ { , . . . , n − } ,j k − n if k ∈ { n, . . . , m + n − } . Once defined the translation on T d , then we are allowed to associate to each site i in T d the shift mapΘ i : S T d → S T d defined by Θ i x = ( x i + j ) j ∈ T d (2.6)at each point x = ( x j ) j ∈ T d . Furthermore, for each k ∈ { , . . . , d − } , we denote by e k the sequence e k = ( k )consisting only of the number k , therefore, the e k ’s are the neighbors of the root o of T d .From now on, we consider the single spin space S = {− , +1 } , so, the state space Ω is described asΩ = {− , +1 } T d . Following [8, 19], we give the definitions of attractive dynamics and of repulsive dynamics.In order to do that we introduce the notation x ≤ y to indicate that x and y are elements of Ω that satisfy x i ≤ y i for all i ∈ T d . 3 efinition 2.3. We call the dynamics P attractive if for every positive integer n , for all configurations x, y such that x ≤ y and each nondecreasing local function f , we have P n ( x, f ) ≤ P n ( y, f ) . (2.7) Definition 2.4.
We call the dynamics P repulsive if for every positive integer n , for all configurations x, y such that x ≤ y and each nondecreasing local function f , we have P n ( x, f ) ≥ P n ( y, f ) . (2.8)By [8, 19] it follows that the dynamics is attractive if and only if for all configurations x, y such that x ≤ y we have p o (+1 | x ) ≤ p o (+1 | y ); furthermore, it is repulsive if and only if for all configurations x, y suchthat x ≤ y we have p o (+1 | x ) ≥ p o (+1 | y ).The PCAs considered in this paper have nondegenerate shift-invariant local transition probabilities suchthat each probabilistic rule p i ( ·| x ) depends only on the spins of the children of i . More precisely, we willstate the following assumptions on the transition kernel. Assumptions: (A1) each p o ( ·| x ) is a probability measure such that p o ( s | x ) > s ∈ {− , +1 } ,(A2) the map x p o ( s | x ) depends only on the values of x on U ( o ) = { e , . . . , e d − } , and(A3) for each i in T d \{ o } , the local transition probability p i ( ·| x ) satisfies p i ( s | x ) = p o ( s | Θ i x ) . (2.9)Note that Assumption (A1) is the so-called nondegeneracy property, while Assumption (A3) is the invarianceof the PCA dynamics under tree shifts. We remark as well that, it follows from ( A
2) and ( A
3) that the map x p i ( s | x ) depends only on the values assumed by the spins of x on U ( i ) = i + { e , . . . , e d − } . One of thecrucial features of this dynamics P is that under Assumptions (A2) and (A3) the relation P n ( x, { y F = ξ } ) = Y i ∈ F P n (Θ i x, { y o = ξ i } ) (2.10)holds for every configuration x , finite volume configuration ( ξ i ) i ∈ F for some F ⊆ T d , and positive integer n . According to Theorem 2.1, every stationary measure for the PCA defined above can be associated to aGibbs measure for the corresponding statistical mechanical model Φ defined by (2.5). Next, we show thatfor the class of PCAs on trees we are dealing with, under suitable conditions, the converse is also valid.
Theorem 2.2.
Under the Assumption (A1)-(A3), let µ be a Gibbs measure for the interaction Φ defined by(2.5), such that it is invariant under time translations, i.e., µ is a Gibbs measure that satisfies µ ( ω m ∈ B ) = µ ( ω m − ∈ B ) for each integer m and each Borel subset B of Ω . Then, there is a stationary measure ν for the correspondingPCA such that µ = µ ν . Therefore, thanks to Theorem 2.2 the study of the ergodicity of the PCA can be closely related to thestudy the uniqueness of the Gibbs measure on space-time associated to it.
Remark.
In Appendix A, we give a more general proof for Theorem 2.2. It actually holds for any PCA on Ω = S V , where S is a nonempty finite set and V is a (locally finite) infinite rooted tree, satisfying ( A and(A2’) Let d : V × V → R be the distance function that assigns to each pair ( i, j ) of vertices the length of theunique path connecting them. Corresponding to each point i that belongs to V the set U ( i ) is a finiteset such that U ( i ) ⊆ { j ∈ V : d ( o , i ) < d ( o , j ) } . (2.11)4 Conditions for ergodicity for PCA on trees
In this section we will present some results regarding sufficient conditions for ergodicity for the classof PCAs described previously. Note that equation (2.10) implies that the probability distributions of thespins at time n are independent, so, this suggests that the typical stationary measures we have to look forare product measures. This remark leads us to state a lemma regarding the characterization of stationaryBernoulli product measures, whose proof is given in Appendix B. Lemma 3.1.
A Bernoulli product measure ν = Bern ( p ) ⊗ T d with parameter p ∈ [0 , , is a stationary measurefor P if and only if Z p o (+1 | x ) ν ( dx ) = p (3.1) i.e. if and only if d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ k = − k ∈ I ( − { m : ξ m = − } p o (+1 | ξ ) p d − l = p . (3.2) Moreover, the probability to find the spin +1 at the root of T d after n + 1 steps of this dynamics startingfrom the configuration x can be written as P n +1 ( x, { y o = +1 } ) (3.3)= d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ k = − k ∈ I ( − { m : ξ m = − } p o (+1 | ξ ) Y k ∈{ ,...,d − }\ I P n (Θ e k x, { y o = +1 } ) . From now on, we will abbreviate +1 by + (resp. − − ). In the first theorem we prove ergodicityresults for the line and the binary trees, while in the second theorem we prove ergodicity and non-ergodicityresults for the 3-ary trees. Theorem 3.1.
Let us consider a PCA with transition probabilities satisfying (A1)-(A3). Then, we havethe following results.(a) If d = 1 , then the PCA dynamics is ergodic. The unique stationary measure is a Bernoulli productmeasure with parameter p = p o (+ |− ) p o ( −| +) + p o (+ |− ) . (3.4) (b) Let d = 2 and the transition probabilities being symmetric under spin-flip, i.e., the equality p o ( s | x ) = p o ( − s | − x ) holds for every spin s and each configuration x . Then the PCA dynamics is ergodic, whereits unique stationary measure is Bern (cid:0) (cid:1) ⊗ T . Theorem 3.2.
Let d = 3 and let the transition probabilities be symmetric under spin-flip. Denote by α := p o (+ | + ++) and γ := p o (+ | − ++) + p o (+ | + − +) + p o (+ | + + − ) . Then the PCA transition rule is(a) ergodic, if α and γ satisfy(i) α − γ = 0 , or(ii) the PCA dynamics is attractive and α − γ = 0 and α + γ ≤ , or(iii) the PCA dynamics is repulsive and α − γ = 0 and α + γ ≥ .In this case the unique stationary measure is given by Bern (cid:0) (cid:1) ⊗ T .(b) non-ergodic, if α and γ satisfy i) α − γ = 0 and α + γ > . In this case, we have several stationary Bernoulli product measureswith parameter p ∈ , q − α )1+ α − γ , q − − α )1+ α − γ , or(ii) the PCA dynamics is repulsive and α − γ = 0 and α + γ < . Remark.
In the last case (Theorem 3.2 ( b ) - ( ii ) ), we can actually prove that the PCA oscillates betweentwo Bernoulli product measures with distinct parameters p . Further details are presented in Section 5.2.3. Before we pass to the proofs of the theorems we will discuss some examples.
For d = 3 and β >
0, let us consider the PCA with transition probabilities given by p i ( s | x ) = 12 s tanh β X k =0 J k x i + e k !! (3.5)where J , J and J ∈ R . Hence, for suitable values of the constants, there exists a critical β c ∈ (0 , ∞ ) suchthat the PCA is ergodic for β ≤ β c and non-ergodic otherwise. In fact the following result holds. Proposition 3.1.
Suppose that one of the following conditions on the coupling constants J , J , J is fulfilled.(C1) J , J , J > and J ≤ J + J , J ≤ J + J , and J ≤ J + J .(C2) J , J , J < and J ≥ J + J , J ≥ J + J , and J ≥ J + J .Let α, γ be defined as in Theorem 3.2, and let function f : R + → R be defined as f ( β ) = 3 α + γ. Then, there exists β c ∈ (0 , ∞ ) depending on the constants J , J , J such that for(a) β ≤ β c the PCA dynamics associated to the local transition probabilities given by (3.5) is ergodic, and(b) β > β c the dynamics is non-ergodic. Remark 1.
Note that, thanks to the spin-flip symmetry of the probabilities (3.5), we can apply Theorem 3.2.Moreover, we remark that the lattice model equivalent to (3.5) has been extensively studied in [6].
Remark 2.
If condition ( C holds, then β c = f − (5) . Otherwise, if ( C holds, then β c = f − (1) .In particular, if J = J = J = J ∈ R \{ } , it follows that β c = | J | log(1 + 2 / ) . In [20] a similarferromagnetic PCA has been studied on Z d where in the particular case d = 2 the value of β c is given by β c = J log(1 + √ . Let us consider the PCA on the 3-ary tree defined as follows. Suppose that at each step every spinassume the value corresponding to the majority among their children. After that each spin make an errorwith a probability (cid:15) ∈ (0 ,
1) independently of each other, that is, if the spin at the site i assumed the value+1 (resp. − − (cid:15) and keep the value +1 (resp. − − (cid:15) . Note that such a system follows a CA dynamics, namely the majority rule, with theaddition of a noise. For a more detailed study of this kind of PCAs, see [27].In the example described above, we have p o (+ | + ++) = p o (+ | + + − ) = p o (+ | + − +) = p o (+ | − ++) = 1 − (cid:15). This PCA has been first studied in [8], where non-ergodicity has been proven only for sufficiently small (cid:15) .In the next proposition we fully characterize its behavior for the whole range of (cid:15) . Proposition 3.2.
There exist two critical values (cid:15) (1) c = and (cid:15) (2) c = such that for every (cid:15) ∈ (0 , (a) the PCA dynamics is ergodic if (cid:15) (1) c ≤ (cid:15) ≤ (cid:15) (2) c , and(b) non-ergodic for (cid:15) / ∈ [ (cid:15) (1) c , (cid:15) (2) c ] . Discussion
In this work we proved the correspondence between stationary measures for PCAs on infinite rootedtrees and time-invariant Gibbs measures for a corresponding statistical mechanical model. As mentionedbefore, the proof of such correspondence is very general and can be applied for any PCA on a (locallyfinite) infinite rooted tree with finite single spin space S . The main implication of this fact is once weestablish conditions for uniqueness of Gibbs measures for such a system, we guarantee the uniqueness ofstationary distributions for the associated PCA, compare also [35]. On the other hand, the existence ofmultiple stationary measures implies on the phase transition in the statistical mechanical model. In thisway we provide a partial relationship between ergodicity and phase transition extending the results from[13].Restricting to the study of PCAs on a d -ary tree T d with translation-invariant local transition probabil-ities with single spin space S = {− , +1 } , we were able to find ergodicity properties for such class of PCAs.The assumption that the choice of a local transition probability at a site i only depends upon the values ofthe spins of the children of i allowed us to derive several important properties, for instance, equations (2.10)and (3.3). Equation (2.10) shows us that the probability distributions of the spins at time n are independent,such fact lead us to characterize the stationary measures of such a system whose form are product measures.In this way, we naturally obtained a polynomial function F defined on the interval [0 ,
1] whose expression isgiven by F ( p ) = d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ k = − k ∈ I ( − { m : ξ m = − } p o (+1 | ξ ) p d − l (4.1)such that, according to equation (3.2), a Bernoulli product measure with parameter p is a stationary measurefor the PCA dynamics if and only if p is a fixed point of F . Furthermore, based on equation (3.3), theconvergence of P n ( x, { y o = +1 } ) for a shift-invariant configuration x (that is, for x that satisfies θ i x = x forall i ) can be studied in terms on the behavior of the iterations F n , since the identity P n +1 ( x, { y o = +1 } ) = F ( P n ( x, { y o = +1 } )) holds.We applied the techniques described above in the cases where d = 1 ,
2, and 3. For d = 1, we the PCAdynamics is ergodic and the unique stationary measure is a Bernoulli product measure with parameter p givenby equation (3.4). Note that this case is equivalent to the study of a PCA on N where the choice of the valueof the spin located at i at time n + 1 depends only on the value of the spin at i + 1 at time n . Extensions ofthis result where U ( i ) = { i, i + 1 } were extensively studied in [8], moreover, more recent generalizations thatconsiders one-dimensional PCAs with general finite alphabets and characterizations of Markov stationarymeasures can be found in [4, 5]. For the cases d = 2 and d = 3 we assumed the invariance of the localtransition probabilities under spin-flip in order to guarantee the existence of a stationary Bernoulli productmeasure (which has parameter ). Under this restriction, we obtained a full characterization the dynamicsof PCAs with d = 2, and d = 3 with the additional hypothesis of attractiveness (resp. repulsiveness).For further generalizations, in order to drop the assumption of spin-flip symmetry and extend the resultsfor any d , it is necessary to investigate the general properties of the polynomial function F regarding itsfixed points and the behavior of its iterates F n . It is also worth investigating generalizations of PCAs fromExamples 1 and 2. Note that Theorem 2.1 together with Dobrushin’s uniqueness theorem implies that fora PCA on T d whose local transition probabilities are given by p i ( s | x ) = 12 s tanh β d − X k =0 J k x i + e k !! (4.2)there is a unique stationary measure given by Bern ( ) ⊗ T d for β small enough, suggesting the ergodicity athigh temperatures.Another kind direction that should be considered in the future is the possibility of inclusion of finitealphabets other that S = {− , +1 } and the possibility of influence of the state at the vertex i at time n onits state at time n + 1, more precisely, the possibility of considering U ( i ) = { i, i + e o , . . . , i + e d − } . Suchassumptions require a new approach once equations (2.10), (3.2) and (3.3) would no longer be valid, so, onepossible direction that should be chosen would be towards an extension of the results from [4, 5].7 Proofs of Ergodicity results
Proof.
Note that a PCA on T is equivalent to a PCA model on Z + . In order to simplify the computations,let us use a and b to denote p o (+ | +) and p o (+ |− ), respectively. Since the local transition probabilities havepositive rates, then, we have | a − b | <
1. It follows that for each point x in Ω , we have P n +1 ( x, { y o = +1 } ) = Z P ( z, { y o = +1 } ) P n ( x, dz )= a · P n ( x, { y e = +1 } ) + b · P n ( x, { y e = − } )= ( a − b ) · P n ( x, { y e = +1 } ) + b = ( a − b ) · P n (Θ e x, { y o = +1 } ) + b for each positive integer n . Note that the relation above can also be obtained by means of equation (3.3).Thus, the quantity above can be expressed as P n ( x, { y o = +1 } ) = ( a − b ) n − · p o (+1 | Θ e + · · · + e | {z } n − x ) + b · n − X k =0 ( a − b ) k . It follows that for any initial configuration x , the probability P n ( x, { y o = +1 } ) converges to p = b − ( a − b ) as n approaches infinity. Therefore, using equation (2.10), we conclude that this PCA is ergodic, where itsunique attractive stationary measure is Bern ( p ) ⊗ T . Proof.
Let a, b ∈ (0 ,
1) defined by a = p o (+ | − − ) = 1 − p o (+ | + +) and b = p o (+ | − +) = 1 − p o (+ | + − ),respectively. Let us show that Bern ( ) ⊗ T , in fact, is the unique attractive stationary measure, that is, forevery initial configuration x we have P n ( x, · ) → Bern ( ) ⊗ T as n approaches infinity. According to equation(3.3), we have P n +1 ( x, { y o = +1 } ) = (1 − b − a ) P n (Θ e x, { y o = +1 } ) + ( b − a ) P n (Θ e x, { y o = +1 } ) + a. By induction, we can show that P n ( x, { y o = +1 } ) = X i ∈{ , } n − (1 − b − a ) { k : i k =0 } ( b − a ) { k : i k =1 } P (Θ i x, { y o = +1 } )+ a n − X l =0 X i ∈{ , } l (1 − b − a ) { k : i k =0 } ( b − a ) { k : i k =1 } . Using the fact that for any real numbers p and q , the relation X i ∈{ , } l p { k : i k =0 } q { k : i k =1 } = ( p + q ) l holds for every nonnegative integer l , it follows that P n ( x, { y o = +1 } ) = X i ∈{ , } n − (1 − b − a ) { k : i k =0 } ( b − a ) { k : i k =1 } P (Θ i x, { y o = +1 } ) + a n − X l =0 (1 − a ) l . (5.1)Since the absolute value of the first term of equation (5.1) is bounded by X i ∈{ , } n − | − b − a | { k : i k =0 } | b − a | { k : i k =1 } = ( | − b − a | + | b − a | ) n − , then lim n →∞ P n ( x, { y o = +1 } ) = a ∞ X l =0 (1 − a ) l = 12 . (5.2)Therefore, by means of equation (2.10), we conclude that Bern ( ) ⊗ T is the unique attractive stationarymeasure of the PCA. 8 .2 Proof of Theorem 3.2 Proof.
Recall we abbreviated α = p o (+ | + ++) and γ = p o (+ | − ++) + p o (+ | + − +) + p o (+ | + + − ). FromLemma 3.1 we know that a stationary product measure has to satisfy the condition Z p o (+1 | x ) ν ( dx ) = p (5.3)which was equivalent to solving equation (3.2), i.e.2(1 + α − γ ) p − α − γ ) p + (3 α − γ − p + (1 − α ) = 0 . (5.4)Since p = is a solution for the equation above, then, it can be written as2 (cid:18) p − (cid:19) (cid:2) (1 + α − γ ) p − (1 + α − γ ) p − (1 − α ) (cid:3) = 0 . (5.5)Suppose that 1 + α − γ = 0. Then, analogously as in the previous case, we have P n ( x, { y o = +1 } )= X i ∈{ , , } n − ( α − p o (+ | + + − )) { k : i k =0 } ( α − p o (+ | + − +)) { k : i k =1 } ( α − p o (+ | − ++)) { k : i k =2 } P (Θ i x, { y o = +1 } )+(1 − α ) n − X l =0 X i ∈{ , , } l ( α − p o (+ | + + − )) { k : i k =0 } ( α − p o (+ | + − +)) { k : i k =1 } ( α − p o (+ | − ++)) { k : i k =2 } . The equation above implies that P n ( x, { y o = +1 } ) → as n approaches infinity, therefore, by means of thesame argument as used in Section 5.1.2, we conclude that the dynamics is ergodic.Now, if 1 + α − γ = 0, we have two other solutions p + = 1 + q − α )1+ α − γ p − = 1 − q − α )1+ α − γ . (5.7)Therefore, both p − and p + are inside the interval (0 ,
1) and are different from if and only if 3 α + γ > Proof.
Let us consider a PCA with attractive dynamics. Again, by using Lemma 3.1, we can find a map F : [0 , → R F ( p ) = 2(1 + α − γ ) p − α − γ ) p + (3 α − γ ) p + (1 − α ) (5.8)such that its fixed points correspond to the parameters of the stationary Bernoulli product measures. Wewill show that F has a unique attractive fixed point at p = , that is, such fixed point satisfies F n ( q ) → p as n approaches infinity for any point q ∈ [0 , F is an increasing function that satisfies F ( p ) > p for all p < ,F ( ) = and F ( p ) < p for all p > . (5.9)Suppose that 1 + α − γ <
0. Due to the attractiveness of the dynamics, it follows that 3 α ≥ γ and theminimum value of F given by F (0) = F (1) = 3 α − γ is nonnegative. Therefore, F is increasing. Moreover,the property (5.9) follows from the identity F ( p ) − p = 2 (cid:18) p − (cid:19) (cid:2) (1 + α − γ ) p − (1 + α − γ ) p − (1 − α ) (cid:3) (5.10)9here (1 + α − γ ) p − (1 + α − γ ) p − (1 − α ) < p = . Now, let us consider the case where1 + α − γ >
0. The attractiveness of the dynamics implies that γ ≥ − α ), so, the minimum value of F is F ( ) = ( − α + γ ) / ≥
0. Again, we prove that F is increasing. Furthermore, we have (5.9) by meansof the equation F ( p ) − p = 2 (cid:18) p − (cid:19) (1 + α − γ )( p − p − )( p − p + ) (5.11)where p − < p + > F is increasing, F (0) = 1 − α < and F (1) = α > , then F ( p ) belongs to (cid:2) − α, (cid:1) ⊆ (cid:2) , (cid:1) for all p in (cid:2) , (cid:1) and F ( p ) belongs to (cid:0) , α (cid:3) ⊆ (cid:0) , (cid:3) for all p in (cid:0) , (cid:3) . Using the continuity of F , we easily conclude thatlim n →∞ F n ( q ) = for every point q that belongs to the interval [0 , p = is the uniqueattractive fixed point for F .It follows from equation (3.3) that P n +1 ( x − , { y o = +1 } ) = F ( P n ( x − , { y o = +1 } ))and P n +1 ( x + , { y o = +1 } ) = F ( P n ( x + , { y o = +1 } )) , where x − and x + are respectively the configurations with all spins − T . The conclusionabove implies that both P n ( x − , { y o = +1 } ) and P n ( x + , { y o = +1 } ) converge to as n approaches infinity.Therefore, since the inequality x − ≤ x ≤ x + holds for every configuration x , it follows from Definition 2.3that P n ( x − , { y o = +1 } ) ≤ P n ( x, { y o = +1 } ) ≤ P n ( x + , { y o = +1 } ) , (5.12)therefore, lim n →∞ P n ( x, { y o = +1 } ) = 12 . (5.13)Finally, we conclude that the probability P n ( x, · ) converges to Bern ( ) ⊗ T as n approaches infinity, inde-pendently on the initial configuration x , hence, the PCA dynamics is ergodic. Proof.
Let us consider a new PCA described by a probability kernel Q defined by Q ( dy | x ) = O i ∈ T q i ( dy i | x ) , (5.14)where each probability q i is given by q i ( · | x ) = p i ( · | − x ) . (5.15)It is easy to see that this PCA satisfies the spin-flip condition. In the case where we have 3 α + γ ≥
1, if weconsider α and γ respectively defined by α = q o (+ | +++) and γ = q o (+ | ++ − )+ q o (+ | + − +)+ q o (+ |− ++),then we have 1 + α − γ = − (1 + α − γ ) = 0 , and 3 α + γ = 6 − (3 α + γ ) ≤ . Therefore, in this case the PCA dynamics described by Q is ergodic. It is easy to check that P n ( x, · ) = Q n (( − n x, · ) holds for every positive integer n and each configuration x . Therefore, the ergodicity of P follows.In order to prove the non-ergodicity for the case 3 α + γ <
1, let us consider again the function F :[0 , → R given by equation (5.8). It is straightforward to show that F ( p ) − (1 − p ) = 2(1 − α − γ ) (cid:18) p − (cid:19) (1 − q − )(1 − q + ) , (5.16)where q − and q + are the elements in the interval (0 ,
1) given by q − = 1 + q − α α − γ q + = 1 − q − α α − γ , (5.18)respectively. It follows that F ( p ) < − p if p ∈ [0 , q − ), F ( p ) > − p if p ∈ ( q − , ), F ( p ) < − p if p ∈ ( , q + ), and F ( p ) > − p if p ∈ ( q + , . (5.19)Because of the repulsiveness of the dynamics, we have 3 α − γ ≤ F ( ) = ( − α + γ ) < −
1, thus, F is a decreasing function. In addition, we have F ( p ) = 1 − F (1 − p ) for every p in [0 , p < F ( p ) < q − if p ∈ [0 , q − ), q − < F ( p ) < p if p ∈ ( q − , ), p < F ( p ) < q + if p ∈ ( , q + ), and q + < F ( p ) < p if p ∈ ( q + , . (5.20)Therefore, we conclude that lim n →∞ F n ( p ) = ( q − if p ∈ [0 , ), and q + if p ∈ ( , n →∞ F n +1 ( p ) = ( q + if p ∈ [0 , ), and q − if p ∈ ( , P n +1 ( x + , · )and P n ( x + , · ) converge to Bern ( q − ) ⊗ T and Bern ( q + ) ⊗ T , respectively, as n approaches infinity. So, thePCA dynamics is not ergodic. Proof.
The PCA is fully described by the numbers p o (+ | + ++) = 12 (1 + tanh β ( J + J + J )) ,p o (+ | + + − ) = 12 (1 + tanh β ( J + J − J )) ,p o (+ | + − +) = 12 (1 + tanh β ( J − J + J )) , and p o (+ | − ++) = 12 (1 + tanh β ( − J + J + J )) . Note that assumption (C1) from Example 1 implies that J + J − J < J + J + J , J − J + J < J + J + J ,and − J + J + J < J + J + J ; and at most one of the quatities J + J − J , J − J + J and − J + J + J can be equal zero. Therefore, the map g : R → R given by g ( β ) = 1 + α − γ = 12 (tanh β ( J + J + J ) − tanh β ( J + J − J ) − tanh β ( J − J + J ) − tanh β ( − J + J + J ))satisfies g (0) = 0 and g ( β ) = 12 J + J + J cosh β ( J + J + J ) − J + J − J cosh β ( J + J − J ) − J − J + J cosh β ( J − J + J ) − − J + J + J cosh β ( − J + J + J ) ! <
12 cosh β ( J + J + J ) (( J + J + J ) − ( J + J − J ) − ( J − J + J ) − ( − J + J + J )) = 0 .
11t follows that g ( β ) = 1 + α − γ < β >
0. Moreover, note that the function f : R → R given by f ( β ) = 3 α + γ = 3 + 32 tanh β ( J + J + J ) + 12 (tanh β ( J + J − J ) + tanh β ( J − J + J ) + tanh β ( − J + J + J ))is increasing, f (0) = 3, and lim β →∞ f ( β ) ≥ . It follows that there is a unique positive real number β c that satisfies f ( β c ) = 5. Since this PCA dynamics satisfies the spin-flip property and is attractive, accordingto Theorem 3.2, the PCA is ergodic for β ≤ β c and non-ergodic for β > β c .Since we proved the result considering the case where condition ( C
1) holds, the proof for the case ( C Proof.
Clearly the PCA satisfies the spin-flip property. Note that in both cases we have 1 + α − γ = 2 (cid:15) − (cid:15) = . Furthermore, note that the PCA is attractive for 0 < (cid:15) < ,repulsive for < (cid:15) <
1, and in both cases we have 1 + α − γ = 0.Let us suppose that (cid:15) ∈ (0 , ). Since 3 α + γ = 6(1 − (cid:15) ), it follows from Theorem 3.2 that the PCA isnon-ergodic for (cid:15) < and ergodic for ≤ (cid:15) < . Now, if (cid:15) ∈ ( , < (cid:15) ≤ and non-ergodic for < (cid:15) < A Appendix
A.1 Proof of Theorem 2.1
Before we follow to the proof of Theorem 2.1 it will be convenient to construct a special sequence (∆ n ) n ∈ N of subsets of Z × V . Given a positive integer n and a nonempty finite subset F of V , let us define a subset∆( n, F ) of Z × V as follows. Let Λ n be the set given byΛ n = { ( n, i ) : i ∈ F } , and for each integer m < n let Λ m = [ x ∈ Λ m +1 U ( x ) ∪ { ( m, i ) : i ∈ F } Then, we define ∆( n, F ) by ∆( n, F ) = n [ m = − n Λ m . Remark.
Observe that ( a ) ∆( n, F ) is a finite subset of Z × V , ( b ) we have {− n, . . . , , . . . , n } × F ⊆ ∆( n, F ) ⊆ {− n, . . . , , . . . , n } × V , and ( c ) for every point x in ∆( n, F ) , if π Z ( x ) = − n , then U ( x ) ⊆ ∆( n, F ) . Now, if ϕ is a one-to-one function from N onto V , then let∆ = ∆(1 , { ϕ (1) } ) , (A.1)and ∆ n +1 = ∆( n + 1 , π V (∆ n ) ∪ { ϕ ( n + 1) } ) (A.2)for each positive integer n . Observe that (∆ n ) n ∈ N is an increasing sequence of elements of S such that Z × V = S n ∈ N ∆ n . Lemma A.1.
Let ∆ = ∆ m for some m ∈ N , and let ∆ be an element of S defined by ∆ = [ x ∈ ∆ π Z ( x )= − m U ( x ) . iven a finite volume configuration ξ in S ∆ , the measure λ ξ on (Ω , F ∆ ) defined by λ ξ ( B ) = Z B Y x =( n,i ) ∈ ∆ p i ( ξ x | ( ξω ∆ c ) n − ) µ ν ( dω ) (A.3) can be expressed as λ ξ ( B ) = Z B [ ξ ] ( ω ) µ ν ( dω ) . (A.4) Proof of Lemma A.1.
It suffices to show the identity for cylinder sets of the form [ ζ ], where each ζ belongsto S ∆ . The result follows by using the fact that the map ω Y x =( n,i ) ∈ ∆ p i ( ξ x | ( ξω ∆ c ) n − )depends only on the values of ω assumed on ∆. Proof of Theorem 2.1.
Let us fix a set Λ ∈ S and a finite volume configuration σ in S Λ . Let ∆ = ∆ m forsome positive integer m such that { x ∈ Z × V : ( { x } ∪ U ( x )) ∩ Λ = ∅} ⊆ ∆ m . Then, for each ω in Ω, we have e − H ΦΛ ( σω Λ c ) = Y x =( n,i )( { x }∪ U ( x )) ∩ Λ = ∅ p i (( σω Λ c ) x | ( σω Λ c ) n − )= Q x =( n,i ) ∈ ∆ p i (( σω Λ c ) x | ( σω Λ c ) n − ) Q x =( n,i ) ∈ ∆( { x }∪ U ( x )) ∩ Λ= ∅ p i ( ω x | ω n − ) , thus e − H ΦΛ ( σω Λ c ) P σ ∈ S Λ e − H ΦΛ ( σ ω Λ c ) = Q x =( n,i ) ∈ ∆ p i (( σω Λ c ) x | ( σω Λ c ) n − ) P σ ∈ S Λ Q x =( n,i ) ∈ ∆ p i (( σ ω Λ c ) x | ( σ ω Λ c ) n − ) . (A.5)Now, given a finite volume configuration η in S ∆ \ Λ , using equation (A.5), we obtain Z [ η ] [ σ ] ( ω ) µ ν ( dω ) = λ ση (Ω) = Z Y x =( n,i ) ∈ ∆ p i (( ση ) x | ( σηω ∆ c ) n − ) µ ν ( dω )= X ζ ∈ S Λ Z e − H ΦΛ ( σηω ∆ c ) P σ ∈ S Λ e − H ΦΛ ( σ ηω ∆ c ) Y x =( n,i ) ∈ ∆ p i (( ζη ) x | ( ζηω ∆ c ) n − ) µ ν ( dω )= X ζ ∈ S Λ Z e − H ΦΛ ( σηω ∆ c ) P σ ∈ S Λ e − H ΦΛ ( σ ηω ∆ c ) λ ζη ( dω )= X ζ ∈ S Λ Z e − H ΦΛ ( σηω ∆ c ) P σ ∈ S Λ e − H ΦΛ ( σ ηω ∆ c ) [ ζη ] µ ν ( dω )= Z [ η ] e − H ΦΛ ( σω Λ c ) P σ ∈ S Λ e − H ΦΛ ( σ ω Λ c ) µ ν ( dω ) . Since (∆ n ) n ∈ N is an increasing sequence of elements of S such that Z × V = S n ∈ N ∆ n , it follows that theequality µ ν ([ σ ] | F Λ c )( ω ) = e − H ΦΛ ( σω Λ c ) P σ ∈ S Λ e − H ΦΛ ( σ ω Λ c ) (A.6)holds for µ ν -almost every point ω in Ω. 13 .2 Proof of Theorem 2.2 Let m and N be integers, where N ≥
0, and let us consider the set∆ = { m } × { j ∈ V : d ( o , j ) ≤ N } . (A.7)If we consider the nonempty finite subset Λ of Z × V given byΛ = N [ l =0 { m + l } × { j ∈ V : d ( o , j ) ≤ N − l } , (A.8)it follows that e − H ΦΛ ( ξω Λ c ) = Y x =( n,i ) ∈ Λ p i ( ξ x | ( ξω Λ c ) n − ) · Y x =( n,i ) / ∈ Λ U ( x ) ∩ Λ = ∅ p i ( ω x | ( ξω Λ c ) n − )= Y x =( n,i ) ∈ Λ p i ( ξ x | ( ξω Λ c ) n − ) · Y x =( n,i ) / ∈ Λ U ( x ) ∩ Λ = ∅ p i ( ω x | ω n − )holds for all finite volume configuration ξ in S Λ and for every ω in Ω. Since µ is a Gibbs measure, then for µ -almost every point ω in Ω we have µ ([ ξ ] | F Λ c )( ω ) = Q x =( n,i ) ∈ Λ p i ( ξ x | ( ξω Λ c ) n − ) P η ∈ S Λ Q x =( n,i ) ∈ Λ p i ( η x | ( ηω Λ c ) n − ) = Y x =( n,i ) ∈ Λ p i ( ξ x | ( ξω Λ c ) n − ) , and summing over all possible spins inside the volume Λ \ ∆, we conclude that µ ([ ξ ∆ ] | F Λ c )( ω ) = Y x =( m,i ) ∈ ∆ p i ( ξ x | ω m − ) . (A.9)If we define the σ -algebra F Proof of Lemma 3.1. Let us proof that given a function a : {− , +1 } d → R and a probability measure µ on {− , +1 } T d , we have X ξ ∈{− , +1 } d a ( ξ ) Y k ∈{ ,...,d − } µ ( x e k = ξ k ) (B.1)= d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ ) Y k ∈{ ,...,d − }\ I µ ( x e k = +1) We prove the equation above by induction. For the case where d = 1, we proof is straightforward. If wesuppose that the result is proven for d , then X ξ ∈{− , +1 } d +1 a ( ξ ) Y k ∈{ ,...,d } µ ( x e k = ξ k )= X ξ ∈{− , +1 } d a ( ξ, +1) Y k ∈{ ,...,d − } µ ( x e k = ξ k ) · µ ( x e d = +1)+ X ξ ∈{− , +1 } d a ( ξ, − Y k ∈{ ,...,d − } µ ( x e k = ξ k ) · µ ( x e d = − X ξ ∈{− , +1 } d a ( ξ, +1) Y k ∈{ ,...,d − } µ ( x e k = ξ k ) · µ ( x e d = +1) − X ξ ∈{− , +1 } d a ( ξ, − Y k ∈{ ,...,d − } µ ( x e k = ξ k ) · µ ( x e d = +1)+ X ξ ∈{− , +1 } d a ( ξ, − Y k ∈{ ,...,d − } µ ( x e k = ξ k )= d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ, +1) Y k ∈{ ,...,d }\ I µ ( x e k = +1) − d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ, − Y k ∈{ ,...,d }\ I µ ( x e k = +1) + d X l =0 ( − l X I ⊆{ ,...,d − }| I | = l X ξ ∈{− , +1 } d ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ, − Y k ∈{ ,...,d − }\ I µ ( x e k = +1) = d X l =0 ( − l X I ⊆{ ,...,d }| I | = l,d/ ∈ I X ξ ∈{− , +1 } d ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ, +1) Y k ∈{ ,...,d }\ I µ ( x e k = +1) − d X l =0 ( − l X I ⊆{ ,...,d }| I | = l,d/ ∈ I X ξ ∈{− , +1 } d ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ, − Y k ∈{ ,...,d }\ I µ ( x e k = +1) d X l =0 ( − l +1 X I ⊆{ ,...,d }| I | = l +1 ,d ∈ I X ξ ∈{− , +1 } d +1 ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ ) Y k ∈{ ,...,d }\ I µ ( x e k = +1) = d X l =0 ( − l X I ⊆{ ,...,d }| I | = l,d/ ∈ I X ξ ∈{− , +1 } d +1 ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ ) Y k ∈{ ,...,d }\ I µ ( x e k = +1) d X l =0 ( − l +1 X I ⊆{ ,...,d }| I | = l +1 ,d ∈ I X ξ ∈{− , +1 } d +1 ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ ) Y k ∈{ ,...,d }\ I µ ( x e k = +1) = d +1 X l =0 ( − l X I ⊆{ ,...,d }| I | = l X ξ ∈{− , +1 } d +1 ξ m = − m ∈ I ( − { m : ξ m = − } a ( ξ ) Y k ∈{ ,...,d }\ I µ ( x e k = +1) . Therefore the result follows.If we consider the the particular case where a ( ξ ) = p o (+1 | ξ ) and µ = Bern ( p ) ⊗ T d that satisfies (3.1),then equation (3.2) follows. Now, if we let a ( ξ ) = p o (+1 | ξ ) and µ = P n ( x, · ), then equations (2.10) and(B.1) implies equation (3.3). References [1] F. Bagnoli, F. Franci, R. Rechtman, Opinion Formation and Phase Transitions in a Probabilistic CellularAutomaton with Two Absorbing Phases , Proceedings 5th International Conference on Cellular Automatafor Research and Industry, Lecture Notes in Computer Science, Springer, 2002.152] R. M. Burton, C.E. Pfister, J. E. Steif, The variational principle for Gibbs states fails on trees , MarkovProcess. Related Fields, , 387–406, 1995.[3] A. Busic, N. Fatès, J. Mairesse, I. Marcovici, Density classification on infinite lattices and trees , Electron.J. Probab. , 2013.[4] J. Casse, Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariantdistribution , Advances in Applied Probability, 48(2), 369-391.[5] J. Casse, J.-F. Marckert, Markovianity of the invariant distribution of probabilistic cellular automata onthe line , Stochastic Processes and their Applications, Volume 125, Issue 9, 2015.[6] P. Dai Pra, P.-Y. Louis, S. Roelly, Stationary measures and phase transition for a class of probabilisticcellular automata , ESAIM Probab. Statist. , 89–104, 2002.[7] B. Derrida, Dynamical phase transition in spin model and automata , Fundamental problem in StatisticalMechanics VII, H. van Beijeren, Editor, Elsevier Science, 1990.[8] R. L. Dobrushin, V. I. Kryukov, A. L. Toom, Locally Interacting Systems and Their Application inBiology , Proceedings of the School-Seminar on Markov Processes in Biology, Held in Pushchino, MoscowRegion, March, 1976.[9] J. Dorrestijn , D. T. Crommelin , J. A. Biello , S. J. Böing, A data-driven multi-cloud model for stochasticparametrization of deep convection , Phil. Trans. A , 2013.[10] P. A. Ferrari, Ergodicity for a class of probabilistic cellular automata , Rev. Mat. Apl., , 93–102, 1991.[11] H.O. Georgii, Gibbs measures and phase transitions , second edition, Walter de Gruyter, Berlin/NewYork, 2011.[12] A. Georges, P. Le Doussal, From equilibrium spin models to probabilistic cellular automata , Journ.Stat. Phys., , 3-4, 1011–1064, 1989.[13] S. Goldstein, R. Kuik, J.L. Lebowitz, and C. Maes, From PCA’s to equilibrium systems and back ,Comm. Math. Phys. , 71–79, 1989.[14] G. Grinstein, C. Jayaprakash, and Y. He, Statistical Mechanics of Probabilistic Cellular Automata Phys. Rev. Lett. , 1985.[15] A.G. Hoekstra, J. Kroc, P.M.A. Sloot, Simulating Complex Systems by Cellular Automata , Springer,2010.[16] J. Kari, Theory of cellular automata: A survey , Theoretical Computer Science, Volume , Issues1-3, 3–33, 2005.[17] L. Lam, Non-Linear Physics for Beginners: Fractals, Chaos, Pattern Formation, Solitons, CellularAutomata and Complex Systems , World Scientific, 1998.[18] J.L. Lebowitz, C. Maes, E.R. Speer, Statistical mechanics of probabilistic cellular automata. , Journ.Stat. Phys. , 1-2, 117–170, 1990.[19] P.-Y. Louis, Automates Cellulaires Probabilistes : mesures stationnaires, mesures de Gibbs associées etergodicité. Université des Sciences et Technologie de Lille - Lille I, 2002.[20] P.-Y. Louis, Ergodicity of PCA: equivalence between spatial and temporal mixing conditions , Electron.Comm. Probab., , 119–134, 2004.[21] P.-Y. Louis, F. R. Nardi, Probabilistic Cellular Automata, Theory, Applications and Future Perspectives .[22] J. Mairesse, I. Marcovici, Probabilistic cellular automata and random fields with i.i.d. directions , AIHPProbabilitès et Statistiques, , 455–475, 2014.[23] J. Mairesse, I. Marcovici, Uniform sampling of subshifts of finite type on grids and trees , Internat. J.Found. Comput. Sci., , 263–287, 2017.[24] C. Maes, S. B. Shlosman, Ergodicity of probabilistic cellular automata: a constructive criterion , Comm.Math. Phys., , 233–251, 1991. 1625] V. A. Malyshev, R. A. Minlos, Gibbs random fields , Kluwer Academic Publishers Group, Dordrecht,1991.[26] D. Makowiec, Modeling Heart Pacemaker Tissue by a Network of Stochastic Oscillatory Cellular Au-tomata , Mauri et al. (editors): Unconventional Computation and Natural Computation, Lecture Notesin Computer Science, , 138–149, 2013.[27] I. Marcovici, M. Sablik, S. Taati, Ergodicity of some classes of cellular automata subject to noise ,preprint arXiv, 2018.[28] J. Palandi, R.M.C. de Almeida, J.R. Iglesias, M. Kiwi, Cellular automaton for the order-disordertransition , Chaos, Solitons & Fractals, Vol. , 439–445, 1995.[29] R. B. Pandey, D. Stauffer, Metastability with probabilistic cellular automata in an HIV infection , Journ.Stat. Phys. , 235–240, 1990.[30] M. Perc, J. Gómez–Gardeñes, A. Szolnoki, L.M. Floría, Y. Moreno, Evolutionary dynamics of groupinteractions on structured populations: A review , J. R. Soc. Interface , 2013.[31] M. Perc and P. Grigolini, Collective behavior and evolutionary games – An introduction , Chaos, Solitons& Fractals , 1–5, 2013.[32] M. Perc and A. Szolnoki, Coevolutionary games – A mini review , Biosystems , 109–125, 2010.[33] A. Procacci, B. Scoppola, E. Scoppola, Probabilistic Cellular Automata for the low-temperature 2d IsingModel , Journal of Statistical Physics, 165, 991-1005 (2016).[34] G. Ch. Sirakoulis and S. Bandini (editors), Cellular Automata: 10th International Conference on Cel-lular Automata for Research and Industry , ACRI 2012, Proceedings, 2012, Lecture Notes in ComputerScience, Springer, 2012.[35] J. Steif, Convergence to equilibrium and spaceâĂŤtime bernoullicity for spin systems in the M < (cid:15) case , Erg. Theory and Dyn. Sys. (3), 547–5775, (1991).[36] T. Tomé, J. R. D. de Felicio, Probabilistic cellular automaton describing a biological immune system , Phys. Rev. E , 3976–3981, 1996.[37] A.L. Toom , N.B. Vasilyev, O.N. Stavskaya, L.G. Mityushin, G.L. Kurdyumov, S.A. Pirogov, Discretelocal Markov systems , in Stochastic Cellular Systems: ergodicity, memory, morphogenesis , edited byR.L. Dobrushin, V.I. Kryukov, A.L. Toom, Manchester University Press, 1–182, 1990.[38] L.N. Vasershtein, Markov processes over denumerable products of spaces describing large system ofautomata , Problemy Peredači Informacii , no. 3, 64–72, (1969).[39] S. Wolfram, Universality and complexity in cellular automata , Physica D: Nonlinear Phenomena, Vol. , Issues 1-2, 1–35, 1984.[40] S. Wolfram, Statistical mechanics of cellular automata , Rev. Mod. Phys., , 3, 601–644, 1983.[41] S. Wolfram, Cellular automata as models of complexity , Nature311