{
  "id": "1503.05614",
  "version": "v1",
  "published": "2015-03-18T23:30:31.000Z",
  "updated": "2015-03-18T23:30:31.000Z",
  "title": "Percolation games, probabilistic cellular automata, and the hard-core model",
  "authors": [
    "Alexander E. Holroyd",
    "Irene Marcovici",
    "James B. Martin"
  ],
  "comment": "17 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $p\\in(0,1)$, and let each site of $Z^2$ be closed with probability $p$ and open with probability $1-p$, independently for different sites. Consider the following two-player game: a token starts at the origin, and a move consists of moving the token 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 one-dimensional probabilistic cellular automaton (PCA), which has already been studied from several perspectives, for example in the enumeration of directed animals in combinatorics, in relation to the golden-mean subshift in symbolic dynamics, and in the context of the hard-core model in statistical physics. The ergodicity of the PCA has been given as an open problem by several authors. Our main result is that the PCA is ergodic for all $p$, and that no draws occur for the game on $Z^2$. A related game, in which the winner is the first player to reach a closed site, is shown to correspond to another PCA. Here we also show that the PCA is ergodic and the game has no draws. There are several natural extensions to dimension $d\\geq 3$, and we conjecture that these games have positive probability of a draw for small $p$; we prove this in a variety of cases using a connection to phase transitions for the hard-core model on appropriately defined $(d-1)$-dimensional lattices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-18T23:30:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C57",
      "60K35",
      "37B15"
    ],
    "keywords": [
      "hard-core model",
      "percolation games",
      "one-dimensional probabilistic cellular automaton",
      "positive probability",
      "ergodicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150305614H"
    }
  }
}