Generalized percolation games on the $2$-dimensional square lattice, and ergodicity of associated probabilistic cellular automata

Abstract

Each vertex of the infinite $2$-dimensional square lattice graph is assigned, independently, a label that reads trap with probability $p$, target with probability $q$, and open with probability $(1-p-q)$, and each edge is assigned, independently, a label that reads trap with probability $r$ and open with probability $(1-r)$. A percolation game is played on this random board, wherein two players take turns to make moves, where a move involves relocating the token from where it is currently located, say $(x,y)\in {\mathbb{Z}}^2$, to one of $(x+1,y)$ and $(x,y+1)$. A player wins if she is able to move the token to a vertex labeled a target, or force her opponent to either move the token to a vertex labeled a trap or along an edge labeled a trap. We seek to find a regime, in terms of $p$, $q$ and $r$, in which the probability of this game resulting in a draw equals $0$. We consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability $r$, target with probability $s$, and open with probability $(1-r-s)$, but the vertices are left unlabeled. Various regimes of values of $r$ and $s$ are explored in which the probability of draw is guaranteed to be $0$. We show that the probability of draw in each such game equals $0$ if and only if a certain probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions to investigate the regimes in which said PCA is ergodic.

84 pages including bibliography. No. of figures: 6. Accepted for publication by the Applied Probability Trust