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

TL;DR

This paper studies one-dimensional probabilistic cellular automata with memory two, analyzing their invariant measures, ergodicity, and reversibility properties, with applications to models in statistical physics and Gibbs random fields.

Contribution

It provides new conditions for invariant measures to be product or Markovian, and extensively explores reversibility phenomena in these automata.

Findings

Invariant measures can be product or Markovian under certain conditions

Ergodicity is established for specific classes of these PCA

Reversibility leads to Gibbs random fields with geometric properties

Abstract

We focus on a family of one-dimensional probabilistic cellular automata with memory two: 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 $t-1$. Such PCA naturally arise in the study of some models coming from statistical physics ($8$-vertex model, directed animals and gaz models, TASEP, etc.). We give conditions for which the invariant measure has a product form or a Markovian form, and we 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.