{
  "id": "1807.10323",
  "version": "v1",
  "published": "2018-07-26T19:03:08.000Z",
  "updated": "2018-07-26T19:03:08.000Z",
  "title": "Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square",
  "authors": [
    "Janko Gravner",
    "David Sivakoff"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $\\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\\mathbb{Z}^2\\times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $\\theta$ exhibits a sharp phase transition, while odd $\\theta$ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $\\mathbb{Z}^2\\times K_n$. The main tool is heterogeneous bootstrap percolation on $\\mathbb{Z}^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-26T19:03:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "two-dimensional lattice",
      "hamming square",
      "gradual percolation transition",
      "sharp phase transition",
      "uniform product measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}