Ergodicity of some probabilistic cellular automata with two letters alphabet via random walks

TL;DR

This paper introduces a novel proof technique for establishing the ergodicity of certain probabilistic cellular automata with two-letter alphabets, using the behavior of boundary regions modeled as random walks.

Contribution

It presents a new proof method based on analyzing boundary dynamics as random walks, applicable to some PCA with errors, advancing understanding of ergodicity in these systems.

Findings

Proves ergodicity for specific PCA with two-letter alphabets.

Uses boundary behavior modeled as random walks to establish ergodicity.

Includes PCA with errors in the analysis.

Abstract

Ergodicity of probabilistic cellular automata is a very important issue in the PCA theory. In particular, the question about the ergodicity of all PCA with two-size neighbourhood, two letters alphabet and positive rates is still open. In this article, we do not try to improve this issue, but we show a new kind of proof (to the best knowledge of the author) about the ergodicity of some of those PCA, including also some CA with errors. The proof is based on the study of the boundaries of islands where the PCA is totally decorrelated from its initial condition. The behaviours of these boundaries are the ones of random walks.