{
  "id": "math/0206132",
  "version": "v1",
  "published": "2002-06-12T21:55:25.000Z",
  "updated": "2002-06-12T21:55:25.000Z",
  "title": "Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation",
  "authors": [
    "Alexander E. Holroyd"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the bootstrap percolation model, sites in an $L$ by $L$ square are initially independently declared active with probability $p$. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as $p \\to 0$ and $L \\to \\infty$ simultaneously of the probability $I(L,p)$ that the entire square is eventually active. We prove that $I(L,p) \\to 1$ if $\\liminf p \\log L > \\lambda$, and $I(L,p) \\to 0$ if $\\limsup p \\log L < \\lambda$, where $\\lambda = \\pi^2/18$. We prove the same behaviour, with the same threshold $\\lambda$, for the probability $J(L,p)$ that a site is active by time $L$ in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold $\\lambda' = \\pi^2/6$. The existence of the thresholds $\\lambda,\\lambda'$ settles a conjecture of Aizenman and Lebowitz, while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-06-12T21:55:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "sharp metastability threshold",
      "two-dimensional bootstrap percolation",
      "probability",
      "values corrects numerical predictions",
      "modified bootstrap percolation model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......6132H"
    }
  }
}