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

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 $0 < p+q \leq 1$ 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.