Percolation games, probabilistic cellular automata, and the hard-core model
Résumé
Let each site of the square lattice $Z^2$ be independently declared closed with probability $p$, and otherwise open. Consider the following game: a token starts at the origin, and the two players take turns to move it from its current site $x$ to an open site in $\{x+(0,1), x+(1,0)\}$; if both these sites are closed, then the player to move loses the game. Is there positive probability that the game is drawn with best play - i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain elementary one-dimensional probabilistic cellular automaton (PCA), which has been studied in the contexts of enumeration of directed animals, the golden-mean subshift, and the hard-core model. Ergodicity of the PCA has been noted as an open problem by several authors.