{
  "id": "1808.03171",
  "version": "v1",
  "published": "2018-08-09T13:59:36.000Z",
  "updated": "2018-08-09T13:59:36.000Z",
  "title": "The speed of critically biased random walk in a one-dimensional percolation model",
  "authors": [
    "Jan-Erik Lübbers",
    "Matthias Meiners"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and H\\\"aggstr\\\"om and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\\mathbb{Z}^d$, namely, for some critical value $\\lambda_{\\mathrm{c}} >0$ of the bias, it holds that the asymptotic linear speed $\\overline{\\mathrm{v}}$ of the walk is strictly positive if the bias $\\lambda$ is strictly smaller than $\\lambda_{\\mathrm{c}}$, whereas $\\overline{\\mathrm{v}}=0$ if $\\lambda \\geq \\lambda_{\\mathrm{c}}$. We show that at the critical bias $\\lambda = \\lambda_{\\mathrm{c}}$, the displacement of the random walk from the origin is of order $n/\\log n$. This is in accordance with simulation results by Dhar and Stauffer for biased random walk on the infinite cluster of supercritical bond percolation on $\\mathbb{Z}^d$. Our result is based on fine estimates for the tails of suitable regeneration times. As a by-product of these estimates we also obtain the order of fluctuations of the walk in the sub-ballistic and in the ballistic, nondiffusive phase.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-09T13:59:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "82B43"
    ],
    "keywords": [
      "one-dimensional percolation model",
      "critically biased random walk",
      "infinite cluster",
      "supercritical bernoulli bond percolation",
      "phase transition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}