Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

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.

21 pages, 0 figure, revised version including a generalization to a larger class of models, structure of the arguments unchanged, minor changes suggested by reviewers, added references