{
  "id": "1706.07338",
  "version": "v1",
  "published": "2017-06-22T14:29:20.000Z",
  "updated": "2017-06-22T14:29:20.000Z",
  "title": "Polluted Bootstrap Percolation in Three Dimensions",
  "authors": [
    "Janko Gravner",
    "Alexander E. Holroyd",
    "David Sivakoff"
  ],
  "comment": "33 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In the polluted bootstrap percolation model, vertices of the cubic lattice $\\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\\to 0$. We show that this density converges to $1$ if $q \\ll p^3(\\log p^{-1})^{-3}$ for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists $C$ such that the final density converges to $0$ if $q > Cp^3$. For the standard model, we establish convergence to $0$ under the stronger condition $q>Cp^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-22T14:29:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "dimensions",
      "polluted bootstrap percolation model",
      "final density converges",
      "occupied neighbor",
      "cubic lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}