{
  "id": "1908.11556",
  "version": "v1",
  "published": "2019-08-30T06:28:12.000Z",
  "updated": "2019-08-30T06:28:12.000Z",
  "title": "Anisotropic bootstrap percolation in three dimensions",
  "authors": [
    "Daniel Blanquicett"
  ],
  "comment": "25 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\\pm e_i$-directions for each $i \\in \\{1,2,3\\}$, where $a_1\\le a_2\\le a_3$. Suppose we infect any healthy vertex $x\\in [L]^3$ already having $a_3+1$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the critical length for percolation up to a constant factor in the exponent, for all triples $(a_1,a_2,a_3)$. To do so, we introduce a new algorithm called the beams process and prove an exponential decay property for a family of subcritical two-dimensional bootstrap processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-30T06:28:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60C05"
    ],
    "keywords": [
      "anisotropic bootstrap percolation",
      "dimensions",
      "subcritical two-dimensional bootstrap processes",
      "infected sites remain infected forever",
      "exponential decay property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}