Invariant Measures and Decay of Correlations for a Class of Ergodic Probabilistic Cellular Automata

TL;DR

This paper establishes new ergodicity conditions for two-state probabilistic cellular automata, analyzes their invariant measures, and demonstrates exponential decay of correlations with explicit constants, advancing understanding of their long-term behavior.

Contribution

It introduces an extended duality approach to derive ergodicity conditions and analyze correlation decay in probabilistic cellular automata.

Findings

Unique invariant measure is shift-mixing in 1D case

Exponential decay of correlations with explicit constants

New sufficient conditions for ergodicity in PCA

Abstract

We give new sufficient ergodicity conditions for two-state probabilistic cellular automata (PCA) of any dimension and any radius. The proof of this result is based on an extended version of the duality concept. Under these assumptions, in the one dimensional case, we study some properties of the unique invariant measure and show that it is shift-mixing. Also, the decay of correlation is studied in detail. In this sense, the extended concept of duality gives exponential decay of correlation and allows to compute explicitily all the constants involved.