{
  "id": "1803.09056",
  "version": "v2",
  "published": "2018-03-24T05:17:59.000Z",
  "updated": "2019-06-11T18:36:33.000Z",
  "title": "A Note on Bootstrap Percolation Thresholds in Plane Tilings using Regular Polygons",
  "authors": [
    "Neal Bushaw",
    "Daniel W. Cranston"
  ],
  "comment": "10 pages, 5 figures; to appear in Australasian J. Combinatorics; this version incorporates reviewer and editor feedback, and differs from the final journal version mainly in typesetting",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "In \\emph{$k$-bootstrap percolation}, we fix $p\\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \\emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings of the plane by regular polygons. A \\emph{vertex type} in such a tiling is the cyclic order of the faces that meet a common vertex. First, we determine the percolation threshold for each of the Archimedean lattices. More generally, let $\\mathcal{T}$ denote the set of plane tilings $T$ by regular polygons such that if $T$ contains one instance of a vertex type, then $T$ contains infinitely many instances of that type. We show that no tiling in $\\mathcal{T}$ has threshold 4 or more. Further, the only tilings in $\\mathcal{T}$ with threshold 3 are four of the Archimedean lattices. Finally, we describe a large subclass of $\\mathcal{T}$ with threshold 2.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2019-06-11T18:36:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "regular polygons",
      "bootstrap percolation thresholds",
      "plane tilings",
      "archimedean lattices",
      "infected faces remain infected forever"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}