{
  "id": "0705.1347",
  "version": "v1",
  "published": "2007-05-09T19:36:32.000Z",
  "updated": "2007-05-09T19:36:32.000Z",
  "title": "Slow Convergence in Bootstrap Percolation",
  "authors": [
    "Janko Gravner",
    "Alexander E. Holroyd"
  ],
  "comment": "22 pages, 3 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-09T19:36:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "slow convergence",
      "bootstrap percolation model",
      "probability",
      "subsequent steps",
      "rigorous explicit bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.1347G"
    }
  }
}