Probabilistic cellular automata and random fields with i.i.d. directions

TL;DR

This paper analyzes one-dimensional probabilistic cellular automata with i.i.d. directions, characterizing conditions for invariant measures and exploring properties of the resulting random fields with weak dependencies and combinatorial structure.

Contribution

It provides necessary and sufficient conditions for Bernoulli product invariant measures and studies the properties of the associated random fields, including extensions to Markovian measures and larger alphabets.

Findings

Characterization of invariant Bernoulli measures for the PCA

The space-time diagram forms a random field with i.i.d. lines in multiple directions

Extensions to Markovian measures and larger alphabets are established

Abstract

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is {0, 1}, and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure. We study the properties of the random field given by the space-time diagram obtained when iterating the PCA starting from its Bernoulli product invariant measure. It is a non-trivial random field with very weak dependences and nice combinatorial properties. In particular, not only the horizontal lines but also the lines in any other direction consist in i.i.d. random variables. We study extensions of the results to Markovian invariant measures, and to PCA with larger alphabets and neighborhoods.