{
  "id": "1705.01652",
  "version": "v1",
  "published": "2017-05-03T23:19:56.000Z",
  "updated": "2017-05-03T23:19:56.000Z",
  "title": "Polluted Bootstrap Percolation with Threshold Two in All Dimensions",
  "authors": [
    "Janko Gravner",
    "Alexander E. Holroyd"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z^d of dimension d>=3 with threshold r=2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d=2 case, where Gravner and McDonald proved that the critical parameter is q/{p^2}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-03T23:19:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "polluted bootstrap percolation model",
      "subsequent steps",
      "cubic lattice",
      "probability",
      "final density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}