逾渗博弈、概率元胞自动机和硬核模型
Percolation games, probabilistic cellular automata, and the hard-core model
摘要：
让正方形晶格$\mathbb{Z}^2$的每个位置独立地被分配三种状态之一：概率为$p$的\textit{trap}，概率为$q$的\textit{target}，概率为$1-p-q$的\textit{open}，其中$0<p+q<1$。考虑以下游戏：代币从原点开始，两个玩家轮流移动，其中移动包括将代币从其当前位置$x$移动到$x+(0,1)$或$x+(1,0)$。将代币移动到陷阱的玩家立即输掉游戏，而将代币移动到目标的玩家立即赢得游戏。游戏是\emph{drawn}且具有最佳玩法的概率是否为正--即\双方都无法取胜这等价于一类初等一维概率元胞自动机（PCA）的遍历性问题。这些自动机已被研究的情况下，枚举的有向格子动物，黄金平均subshift，和硬核模型，其遍历性已被注意到作为一个开放的问题，由几位作者。我们证明了这些PCA是遍历的，相应地，$\mathbb{Z}^2$上的游戏没有平局。另一方面，我们证明了某些类似的游戏\emph{do}展示绘制适当的参数值在各种有向图在更高的维度，包括定向版本的甚至子格的$\mathbb{Z}^d$在所有$d\geq3$。这是证明通过一个硬核晶格气尺寸$d-1$的降维。我们表明，绘制发生时，相应的硬核模型有多个吉布斯分布。我们猜想，提请也发生在标准定向格$\mathbb{Z}^d$的$d\geq 3$，但在这里，我们的方法遇到了一个根本性的障碍。
英文摘要：
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 $d\geq3$. 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 $d\geq 3$, but here our method encounters a fundamental obstacle.
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