Bond percolation games and their connections with probabilistic cellular automata

Abstract

We consider \emph{bond percolation games} on the $2$-dimensional square lattice in which each edge (that is either between the sites $(x,y)$ and $(x+1,y)$ or between the sites $(x,y)$ and $(x,y+1)$, for all $(x,y) \in \mathbb{Z}^{2}$) has been assigned, \emph{independently}, a label that reads \emph{trap} with probability $p$, \emph{target} with probability $q$, and \emph{open} with probability $1-p-q$. Once a realization of this labeling is generated, it is revealed in its entirety to the players before the game starts. The game involves a single token, initially placed at the origin, and two players who take turns to make \emph{moves}. A \emph{move} involves relocating the token from where it is currently located, say the site $(x,y)$, to any one of $(x+1,y)$ and$(x,y+1)$. A player wins if she is able to move the token along an edge labeled a target, or if she is able to force her opponent to move the token along an edge labeled a trap. The game is said to result in a draw if it cont...