{
  "id": "1311.5369",
  "version": "v2",
  "published": "2013-11-21T11:21:35.000Z",
  "updated": "2015-06-19T07:30:38.000Z",
  "title": "Branching random walks and multi-type contact-processes on the percolation cluster of ${\\mathbb{Z}}^{d}$",
  "authors": [
    "Daniela Bertacchi",
    "Fabio Zucca"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1040 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 4, 1993-2012",
  "doi": "10.1214/14-AAP1040",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{\\mathbb{Z}}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${{\\mathcal{C}}_{\\infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${{\\mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{\\mathcal{C}}_{\\infty}}$. This implies that the strong critical parameters of the branching random walk on ${{\\mathbb{Z}}^d}$ and on ${{\\mathcal{C}}_{\\infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-21T11:21:35.000Z",
      "title": "Branching random walks and multi-type contact-processes on the percolation cluster of $\\mathbb{Z}^d$",
      "abstract": "In this paper we prove that under the assumption of quasi-transitivity, if a branching random walk on $\\mathbb{Z}^d$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster $\\mathcal{C}_\\infty$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on $\\mathbb{Z}^d$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on $\\mathcal{C}_\\infty$. This implies that the strong critical parameters of the branching random walk on $\\mathbb{Z}^d$ and on $\\mathcal{C}_\\infty$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.",
      "comment": "13 pages, 6 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-19T07:30:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K37"
    ],
    "keywords": [
      "branching random walk",
      "multi-type contact-processes",
      "strong critical parameters",
      "type contact process survives",
      "individuals"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.5369B"
    }
  }
}