Probabilistic cellular automata with memory two: invariant laws and multidirectional reversibility

Casse, Jérôme; Marcovici, Irène

Annales Henri Lebesgue, Tome 3 (2020), pp. 501-559.

Considérons la famille d’automates cellulaires probabilistes (ACP) de dimension un avec mémoire deux ayant la propriété suivante : la dynamique est telle que la valeur d’une cellule au temps $t+1$ est tirée aléatoirement selon une distribution qui est une fonction de l’état de ses deux voisines les plus proches au temps $t$, et de son propre état au temps $t-1$. Nous donnons des conditions pour lesquelles la loi invariante d’un tel ACP est une mesure de forme produit ou une mesure markovienne, et prouvons un résultat d’ergodicité s’appliquant dans ce contexte. Les diagrammes espace-temps de ces ACP possèdent différentes formes de réversibilité. Nous décrivons et étudions ce phénomène, qui fournit des familles de champs aléatoires de Gibbs sur la grille carrée ayant des propriétés géométriques et combinatoires remarquables. De tels ACP apparaissent de manière naturelle dans l’étude de différents modèles de physique statistique. En utilisant le point de vue des ACP, nous retrouvons des résultats portant sur le modèle à $8$ sommets et sur l’énumération des animaux dirigés, et nous montrons aussi que nos méthodes permettent de trouver de nouveaux résultats sur une extension du modèle classique de TASEP. Un autre résultat original de ce travail est la description de familles d’ACP pour lesquels la loi invariante est explicite, mais n’est ni une mesure de forme produit, ni une mesure markovienne.

Let us consider the family of one-dimensional probabilistic cellular automata (PCA) with memory two having the following property: the dynamics is such that the value of a given cell at time $t+1$ is drawn according to a distribution which is a function of the states of its two nearest neighbours at time $t$, and of its own state at time $t-1$. We give conditions for which the invariant measure has a product form or a Markovian form, and prove an ergodicity result holding in that context. The stationary space-time diagrams of these PCA present different forms of reversibility. We describe and study extensively this phenomenon, which provides families of Gibbs random fields on the square lattice having nice geometric and combinatorial properties. Such PCA naturally arise in the study of different models coming from statistical physics. We review from a PCA approach some results on the $8$-vertex model and on the enumeration of directed animals, and we also show that our methods allow to find new results for an extension of the classical TASEP model. As another original result, we describe some families of PCA for which the invariant measure can be explicitly computed, although it does not have a simple product or Markovian form.

Classification : 60J05, 60G10, 60G60, 37B15, 37A60
Mots clés : Probabilistic cellular automata, invariant measures, ergodicity, reversibility

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