Ergodicity of a generalized probabilistic cellular automaton with parity-based neighbourhoods

Abstract

We study a one-dimensional generalized probabilistic cellular automaton $E_{p,q}$ with universe $\mathbb{Z}$, alphabet $\mathcal{A}=\{0,1\}$, parameters $p$ and $q$ such that $0<p+q\le 1$ and two neighbourhoods ${\mathcal{N}}_0=\{0,1\}$ and $\mathcal{N}=\{1,2\}$. The state $E_{p,q}\eta (x)$ of any $x\in \mathbb{Z}$ under the application of $E_{p,q}$ is a random variable whose probability distribution depends on the states $\eta (x+y)$ for $y\in {\mathcal{N}}_i$ where $i$ has the same parity as $x$. We establish ergodicity of this GPCA for various ranges of values of $p$ and $q$ via its connection with a suitable percolation game on a two-dimensional lattice. For these same ranges of values of $p$ and $q$, we show that the above-mentioned game has probability $0$ of resulting in a draw.

arXiv comment20 pages, 3 figures