{
  "id": "math/0603645",
  "version": "v2",
  "published": "2006-03-28T09:01:10.000Z",
  "updated": "2006-04-03T20:29:58.000Z",
  "title": "The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions",
  "authors": [
    "Alexander E. Holroyd"
  ],
  "comment": "20 pages, 3 figures (added discussion, corrected typo in (24))",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\\to\\infty and p\\to 0 simultaneously, this probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to 0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n denotes the n-th iterate of the exponential function, and the threshold lambda equals pi^2/6 for all d.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-04-03T20:29:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "metastability threshold",
      "dimensions",
      "threshold lambda equals",
      "active site remains active forever",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3645H"
    }
  }
}