On a class of probabilistic cellular automata with size-$3$ neighbourhood and their applications in percolation games

TL;DR

This paper links the ergodicity of a class of size-3 neighborhood probabilistic cellular automata to the outcome probabilities of percolation games on a 2D lattice, establishing conditions for the absence of draws.

Contribution

It demonstrates that the ergodicity of a specific class of probabilistic cellular automata determines the likelihood of draws in related percolation games, providing a rigorous connection.

Findings

Probability of draws is zero when p+q > 0.

Ergodicity of the cellular automaton is equivalent to zero draw probability.

Established ergodicity for a broad class of PCAs.

Abstract

Different versions of percolation games on $\mathbb{Z}^{2}$, with parameters $p$ and $q$ that indicate, respectively, the probability with which a site in $\mathbb{Z}^{2}$ is labeled a trap and the probability with which it is labeled a target, are shown to have probability $0$ of culminating in draws when $p+q > 0$. We show that, for fixed $p$ and $q$, the probability of draw in each of these games is $0$ if and only if a certain $1$-dimensional probabilistic cellular automaton (PCA) $F_{p,q}$ with a size-$3$ neighbourhood is ergodic. This allows us to conclude that $F_{p,q}$ is ergodic whenever $p+q > 0$, thereby rigorously establishing ergodicity for a considerable class of PCAs.