Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

TL;DR

This paper proves that certain probabilistic cellular automata, including Toom's model, exhibit exponential decay of correlations and strong mixing properties in the low-noise regime, indicating rapid convergence to a stable state.

Contribution

It establishes exponential decay of correlations and mixing for a broad class of monotonic probabilistic cellular automata, including Toom's model, in the low-noise regime.

Findings

Exponential convergence to an extremal invariant measure.

Exponential decay of correlations in space and time.

Strong mixing properties of the invariant measure.

Abstract

We investigate the low-noise regime of a large class of probabilistic cellular automata, including the North-East-Center model of Toom. They are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion. We prove, for a set of initial conditions, exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value. We also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing.