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- W2118009076 abstract "Let each site of the square lattice $mathbb{Z}^2$ be independently assigned one of three states: a textit{trap} with probability $p$, a textit{target} with probability $q$, and textit{open} with probability $1-p-q$, where $0<p+q<1$. Consider the following game: a token starts at the origin, and two players take turns to move, where a move consists of moving the token from its current site $x$ to either $x+(0,1)$ or $x+(1,0)$. A player who moves the token to a trap loses the game immediately, while a player who moves the token to a target wins the game immediately. Is there positive probability that the game is emph{drawn} with best play -- i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain family of elementary one-dimensional probabilistic cellular automata (PCA). These automata have been studied in the contexts of enumeration of directed lattice animals, the golden-mean subshift, and the hard-core model, and their ergodicity has been noted as an open problem by several authors. We prove that these PCA are ergodic, and correspondingly that the game on $mathbb{Z}^2$ has no draws. On the other hand, we prove that certain analogous games emph{do} exhibit draws for suitable parameter values on various directed graphs in higher dimensions, including an oriented version of the even sublattice of $mathbb{Z}^d$ in all $dgeq3$. This is proved via a dimension reduction to a hard-core lattice gas in dimension $d-1$. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice $mathbb{Z}^d$ for $dgeq 3$, but here our method encounters a fundamental obstacle." @default.
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- W2118009076 date "2015-03-18" @default.
- W2118009076 modified "2023-09-27" @default.
- W2118009076 title "Percolation games, probabilistic cellular automata, and the hard-core model" @default.
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