{
  "id": "math/0410465",
  "version": "v1",
  "published": "2004-10-21T13:59:41.000Z",
  "updated": "2004-10-21T13:59:41.000Z",
  "title": "Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation",
  "authors": [
    "Federico Camia"
  ],
  "comment": "15 pages",
  "doi": "10.1007/s10955-004-8778-4",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a cellular automaton with state space $\\{0,1 \\}^{{\\mathbb Z}^2}$ where the initial configuration $\\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $\\omega_n$ at time n converges exponentially fast to a final configuration $\\bar\\omega$, and that the limiting measure corresponding to $\\bar\\omega$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $\\beta$, $\\eta$, $\\nu$ and $\\gamma$, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ${\\mathbb Z}^2$ (i.e., for independent $*$-percolation on ${\\mathbb Z}^2$), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents. This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-21T13:59:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B27",
      "60K35",
      "82C22",
      "82B43",
      "82C20",
      "82C43",
      "37B15"
    ],
    "keywords": [
      "two-dimensional bootstrap percolation",
      "critical exponents",
      "scaling limit",
      "cellular automaton",
      "bernoulli product measure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Physics",
      "year": 2005,
      "month": "Jan",
      "volume": 118,
      "number": "1-2",
      "pages": 85
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005JSP...118...85C"
    }
  }
}