Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

For a general attractive Probabilistic Cellular Automata on S Z d , we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the inuence from the boundary for the invariant measures of the system restricted to nite boxes. For a class of reversible PCA dynamics on {--1, +1} Z d , with a naturally associated Gibbsian potential $\varphi$, we prove that a (spatial-) weak mixing condition (WM) for $\varphi$ implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to $\varphi$ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.


Introduction
The main feature of Probabilistic Cellular Automata dynamics (usually abbreviated in PCA) is the parallel, or synchronous, evolution of all interacting elementary components. They are precisely discrete-time Markov chains on a product space S Λ (configuration space) whose transition probability is a product measure. In this paper, S (spin space) is assumed to be a finite set with total order denoted by and Λ (set of sites) a subset, finite or infinite, of Z d . The fact that the transition probability kernel P (dσ|σ ′ ) (σ, σ ′ ∈ S Λ ) is a product measure means that all spins {σ k : k ∈ Λ} are simultaneously and independently updated. This transition mechanism differs from the one in the most common Gibbs samplers, where Electronic Communications in Probability only one site is updated at each time step. In opposition to these dynamics with sequential updating, it is simple to define PCA's on the infinite set S Z d without passing to continuous time.
The main purpose of this article is to study relation between different types of conditions which insure the fastest convergence towards an equilibrium state (νP = ν) of PCA dynamics on S Z d . Let us emphasise that the non-degeneracy hypothesis we will assume implies that the asymptotic behaviour of PCA dynamics on S Λ where Λ is a finite subset of Z d (called finite volume PCA dynamics) is well-known. It is a classical result from the theory of finite state space aperiodic irreducible Markov Chains. Such discrete time processes admit a unique stationary probability measure, and are ergodic. However, if the PCA dynamics is considered on S Z d (infinite volume dynamics), some non-ergodic behaviour may arise (see for instance example 2 section III in [8]). The most famous condition which insures ergodicity of the PCA dynamics on S Z d is due to Dobrushin and Vasershtein's work (see [15]), and applies in the high-temperature regime. Others conditions of ergodicity for general PCA can be found in the following works: [4,7,9,12,13]. See for instance Sections 6.1.2 and 6.1.3 in [10] for details. They all are effective only when some high-temperature condition holds or in perturbative cases.
We will here adopt another approach, partially inspired by Martinelli and Olivieri's work for a class of continuous time Interacting Particle Systems called Glauber dynamics (see [14]), and based on a famous statement of Holley about rate of convergence ( [6]). We introduce a condition (A) which means the exponential decay of the influence from the boundary for the invariant measure of the system restricted to any finite box, which will be here proved to be equivalent to the exponentially fast ergodicity (Theorem 1). The condition (A) we use is not a constructive criterion like the Dobrushin-Vasershtein condition, or its generalised version developed in [12] and numerically studied in [2]. But, theoretically, comparison of spatial and time mixing are always interesting (cf. [14,3]). Furthermore we present different examples in which (A) is satisfied on a larger domain than Dobrushin-Vasershtein condition, and is moreover optimal for these models.
In section 2 we state our main results. The first and more general one (Theorem 1) is the following: convergence towards equilibrium in the uniform norm with an exponential rate is equivalent to the condition (A). In other words exponential mixing in space is equivalent to exponential mixing in time. It will then be applied to a class of reversible PCA dynamics on {−1, +1} Z d , associated in a natural way to a Gibbsian potential ϕ. We prove that the usual weak mixing condition for ϕ implies the validity of (A), thus the exponential ergodicity of the dynamics towards the unique Gibbs measure associated to ϕ holds (Theorem 2). For some particular PCA of this class, we also prove that (A) is weaker than the Dobrushin-Vasershtein ergodicity condition and note that the exponential ergodicity holds as soon as there is no phase transition. Our result are then the first optimal ones in this context. Sections 3 and 4 are respectively devoted to the proof of the Theorems and useful Lemmas.

Main results
Let P denotes a PCA dynamics on S Z d . This means a Markov Chain on S Z d whose transition probability kernel P verifies for all configuration η ∈ S Z d , where for all site k ∈ Z d , for all η, p k ( . |η) is a probability measure on S, called updating rule. For any subset ∆ of Z d , and for all configurations σ and η of S Z d , the configuration σ ∆ η ∆ c is defined by σ k if k ∈ ∆, else η k . Let the notation σ ∆ design (σ k ) k∈∆ too. Let Λ be a finite subset of Z d (denoted by Λ ⋐ Z d ). We call finite volume PCA dynamics with boundary condition τ (τ ∈ S Z d or τ ∈ S Λ c ), the Markov Chain on S Λ whose transition It may be identified with the following infinite volume PCA dynamics on S Z d : . Let ν τ Λ denote the stationary measure associated to the finite volume dynamics P τ Λ . For ν probability measure on S Z d (equipped with the Borel σ-field associated to the product topology), νP refers to νP (dσ) = P (dσ|η)ν(dη). Recursively νP (n) = (νP (n−1) )P .
For each function f on S Z d , P (f ) is the function defined by P (f )(η) = f (σ)P (dσ|η). All the measures considered in this paper are probability measures.
PCA dynamics considered here are assumed to be non degenerate: ∀k ∈ Z d , ∀η ∈ S Z d , ∀s ∈ S, p k ( s | η ) > 0; they are also local, which means: Attractivity of PCA dynamics is moreover assumed here: One can order two configurations by defining σ η if ∀k ∈ Λ, σ k η k . A real function f on S Λ will then be said to be increasing if σ η implies f (σ) f (η). Thus two probability measures ν 1 and ν 2 satisfy the stochastic ordering ν 1 ν 2 if, for all increasing functions f on S Λ , ν 1 (f ) ν 2 (f ), with the notation ν i (f ) = f (σ)ν i (dσ). As Markov chain, a PCA dynamics P on S Λ (Λ ⊂ Z d ) is attractive if for all increasing function f , P (f ) is still increasing. Let us define too, for s ∈ S, σ ∈ S Λ , the function G k (s, σ) by: Recall that a PCA dynamics is attractive if, and only if, for all k in Λ, and all value s ∈ S, the function G k (s, .) is increasing (in σ).
We define, for each f continuous function on the compact S Z d and for all k in Z d , Theorem 1 Let S be a totally ordered finite set with maximal (resp. minimal) element denoted by +(resp. −). + + + (resp. − − −) denotes configurations equal to + (resp. −) in all sites. Let P be an attractive, translation invariant, non degenerate, local PCA dynamics on S Z d . Let ν + + +

B(L)
(resp. ν − − − B(L) ) be the stationary measure of P + + + B(L) (resp. P − − − B(L) ). The following spatial mixing condition: ∃C > 0, ∃M > 0, ∃L 1 ∈ N * , ∀L ∈ N * , L L 1 , is equivalent to the convergence of the dynamics P towards the unique equilibrium state ν with exponential rate: ∃λ > 0, ∃n 1 , ∀n n 1 , ∀f local function on S Z d : In order to better interpret the meaning of condition (A) and the relevance of Theorem 1, we then apply it to a wide class of reversible PCA dynamics on {−1, +1} Z d . First, let us recall some known facts about reversible PCA dynamics (that is to say PCA dynamics whose set of reversible measures R is not empty). The study of the qualitative nature of their equilibrium states as Gibbs measures was initiated by Kozlov and Vasilyev (see [8,16] Assume until the end of this section and in section 4 that where β is a positive real parameter and K : Z d → R is an interaction function between sites which is symmetric and has finite range R > 0 (i.e. for all k of Z d such that k 1 > R then K(k) = 0). Remark that β = 0 is the independent case (sites don't interact), and that when β increases, the dynamics becomes less and less random. So β may be thought as a kind of inverse temperature parameter. See subsection 4.1.1 in [10] for the generality of the class C 0 among reversible PCA dynamics on {−1, +1} Z d . Due to their definition, PCA dynamics in C 0 are local, translation invariant, non degenerate. It is known (see [8,1]) that any PCA dynamics P in C 0 admits at least one reversible measure which is a Gibbs measure associated to the following translation invariant multibody potential ϕ: Moreover Proposition 3.3 in [1] stated the precise relations R = S ∩ G(ϕ) and R s = S s , where S (resp. R) denotes the set of P -stationary (resp. P -reversible) measures, S s and R s their respective space-translation invariant measures' parts, and G(ϕ) the set of Gibbs measures on S Z d associated to the potential ϕ.
One also checks that such a PCA dynamics P is attractive, if and only if function K(.) is non-negative (see Property 4.1.2 in [10]). From now on, let us assume that K is non negative.
Mixing conditions for a potential ϕ define different regions in the domain of absence of phase transition for the associated Gibbs measures. Strong mixing conditions are usually related to the domain where Dobrushin's uniqueness holds, and weak mixing conditions are expected to be valid in the main part of the uniqueness domain: See [14] for a review on these conditions. Here, we call weak mixing condition for the potential ϕ, the condition: where µ is the unique Gibbs measure associated to ϕ. For ferromagnetic potentials, it is indeed the equivalent form of more general weak mixing condition.
Theorem 2 Let P be an attractive PCA dynamics on {−1, +1} Z d of the class C 0 defined by (3), let ϕ denote the potential canonically associated defined in (4), and G(ϕ) the set of Gibbs measures w.r.t ϕ.
• Otherwise, when there is no phase transition (i.e. G(ϕ) = {µ}) the dynamics P is ergodic towards the unique Gibbs measure µ. Moreover if we assume the potential ϕ satisfies the weak mixing condition (WM), then the convergence towards µ holds with exponential rate.
In [1], we established that, for nearest neighbour interaction function K, phase transition holds for β large. For instance, when d = 2, let P J be the PCA dynamics of the class C 0 obtained taking: is a basis of R 2 and J a positive constant. The canonically associated potential ϕ J (cf. (4) ) is the following four-body potential: From Theorem 2 we conclude here that for β large, the PCA P J is non-ergodic since it has at least two different stationary states ν − and ν + .
Let us now discuss how large is the domain where condition (WM) holds. One conjectures Weak Mixing condition for Gibbs measure is valid up to the critical temperature, that is, as soon as there is no phase transition. In that sense, our main result would give ergodicity with exponential rate on a much larger region as the region where the Dobrushin-Vasershtein criterion holds. In fact, let us mention the reference [5], where, using percolation techniques, it is proved that in dimension d = 2, for a ferromagnetic nearest neighbour Ising model without extremal magnetic field, the associated Gibbs measure is weak mixing as soon as it is unique (i.e. ∀β, β < β c ). In order to precise this assertion, let us consider the dynamics P J . A projection argument relates the potential ϕ J associated to P J with the usual Ising ferromagnetic pair potential with intensity coefficient J (see [16]). Due to Higuchi's result, we know that the Gibbs state associated to this potential ϕ J is weak mixing as soon as there is no phase transition, which happens for β lower than the critical value β c , which coincides with the Ising critical inverse temperature β c = log(1+ . In other words, we obtain that the PCA dynamics P J is ergodic with exponential rate for β < β c and non-ergodic for β > β c . Taking J = 1, β c ≃ 0.441; since Dobrushin-Vasershtein criterion applies only for β < 1 2 Argth( 1 2 ) ≃ 0.275 (cf. part 6.1.2 in [10]), ours is better.

Proof of the Theorem 1
The proof of Theorem 1 is based on the existence of some coupling of PCA dynamics preserving the stochastic ordering. Let (P 1 , P 2 , . . . , P N ) be an increasing N -uple of PCA dynamics which means PCA related by the following monotonicity property ∀k ∈ Z d , ∀ζ 1 where G i is the function associated to P i by (1). There exists (cf. [11]) a monotone synchronous coupling on (S Z d ) N denoted by P 1 ⊛ P 2 ⊛ . . . ⊛ P N with the following property: for all initial configuration σ 1 σ 2 . . . σ N and for all times n, Such a coupling will be called increasing synchronous coupling. The notation IP denotes the coupling P ⊛ P ⊛ . . . ⊛ P of N times the same PCA dynamics P , where N will be a finite large enough number.
This coupling allows us to develop some monotonicity argument and to state the following result, whose proof is in [11]: is the maximal (resp. minimal) measure of the set {ν τ Λ : τ ∈ S Λ c } of stationary measures associated to the PCA dynamics P τ Λ on the fixed finite volume Λ and with boundary condition τ . Let ν + + + and ν − − − denote the maximal and the minimal elements of the set S of stationary measures associated to the PCA dynamics P . Following relations hold: In particular, P admits a unique stationary measure ν if and only if ν − − − = ν + + + .
Note that P (n) denotes P • P • . . . • P , and so is for instance δ + + + P (n) the law at time n of the Markov Chain with transition kernel P and initial distribution δ + + + .

Proof. ((A) implies (2) in Theorem 1)
The most delicate part is to establish the exponential rate of convergence towards equilibrium. Our proof is inspired by Martinelli and Olivieri proof of exponential ergodicity for continuous time Glauber dynamics on {−1, +1} Z d (see [14]). For any time n ∈ N, let us define a coefficient which controls the ergodicity: If we assume the exponential bound (A), thanks to forthcoming Lemma 8, we deduce that lim n→∞ ρ(n) = 0. Reporting assumption (A) in the inequality (10), we can use forthcoming Lemma 11 to deduce that (ρ(n)) n∈N * converge to 0 faster than 1 n d . Finally, using inequality (9) and Lemma 12, we conclude that ρ(n) converges to 0 exponentially fast; thus, thanks to Lemma 7, conclusion holds. ✷ Technical lemmas: First remark the easy fact: Lemma 5 Let (Ω, A, P) be a probability space, and Z a random variable with values in a finite set {z 1 < . . . < z m } of R, such that P(Z 0) = 1. Then, if κ = max{ 1 zi , z i > 0, 1 i m} and κ ′ = max{z i , 1 i m} (which do not depend on the law of Z under P) we have: P(Z = 0) κ ZdP and ZdP κ ′ P(Z = 0).
Using the monotonicity property of the coupling, the two following Lemmas are easily proved.

Electronic Communications in Probability
Note that due to the monotonicity of ρ(.), we can restrict ourselves to the case ρ(.) > 0.

Remark 9
As an immediate consequence of Lemma 8 we get lim n→∞ ρ(n) = 0, which implies the ergodicity of P thanks to Lemma 7.
Let us denote by R = max k ′ ∈V0 k ′ 1 the finite range of the local translation invariant PCA dynamics P .
Analogously we prove (b) σ 0 dν − − − B(L) . Thus, the following inequality holds: which gives the estimate of the second term in inequality (11). The first term is treated in the same way. So the recursive inequality (10) is established. ✷ We now state some general analytic lemmas; for proofs see [10,14].

Proof of the Theorem 2
For general PCA in finite volume, invariant measures are not explicitly known; but for the class C 0 here considered, we computed them (cf. Proposition 3.1 in [1]). The unique reversible measure for the PCA dynamics P τ Λ is defined by whereσ = σ Λ τ Λ c , and W τ Λ is the normalisation factor. Such measure does not coincide with the finite volume Gibbs measures µ τ Λ (σ) = 1 Z τ Λ exp(− A⊂Z d ,A∩Λ =∅ ϕ A (σ Λ τ Λ c )) contrary to what happens for Glauber dynamics when detailed balance holds. Nevertheless, they are related as relation (18) attempts. We will not write down all technical computations which prove relations (18), (19). Interested reader may refer respectively to Proposition 4.1.8 and Property 4.1.12 in [10].
Let Λ, Λ ′ two finite subsets of Z d such that Λ ⊂ Λ ′ and ∂ i Λ ∩ ∂ i Λ ′ = ∅, where ∂ i Λ {k ∈ Λ : U k ∩ Λ c = ∅}. Let τ ′ be a boundary condition of Λ and µ τ ′ Λ denotes the finite volume Gibbs distribution associated to the potential ϕ on the volume Λ with boundary condition τ ′ . We then state: Note that the potential ϕ is not really a ferromagnetic potential in the usual sense. However we can check that associated finite volume Gibbs measures verify a kind of monotone behaviour: τ 1 τ 2 ⇒ µ τ1 are extremal states in the sense of stochastic ordering of the set G(ϕ). Recall µ probability measure on S Z d is in G(ϕ) if, per definitionem, for any finite volume Λ ⊂ Z d , a version of the conditioned measure µ(dσ Λ |σ Λ c ) is µ σ Λ c Λ (dσ Λ ). Finally, let us state the following lemma: Lemma 13 If the Weak Mixing Condition (WM) holds for the potential ϕ associated to the PCA dynamics P , then assumption (A) holds for P .