{
  "id": "1705.00671",
  "version": "v1",
  "published": "2017-05-01T19:17:08.000Z",
  "updated": "2017-05-01T19:17:08.000Z",
  "title": "Regularity of the speed of biased random walk in a one-dimensional percolation model",
  "authors": [
    "Nina Gantert",
    "Matthias Meiners",
    "Sebastian Mueller"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelsson-Fisk and H\\\"aggstr\\\"om established for this model a phase transition for the asymptotic linear speed $\\overline{\\mathrm{v}}$ of the walk. Namely, there exists some critical value $\\lambda_{\\mathrm{c}}>0$ such that $\\overline{\\mathrm{v}}>0$ if $\\lambda\\in (0,\\lambda_{\\mathrm{c}})$ and $\\overline{\\mathrm{v}}=0$ if $\\lambda>\\lambda_{\\mathrm{c}}$. We show that the speed $\\overline{\\mathrm{v}}$ is continuous in $\\lambda$ on the interval $(0,\\lambda_{\\mathrm{c}})$ and differentiable on $(0,\\lambda_{\\mathrm{c}}/2)$. Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of $\\overline{\\mathrm{v}}$ on $(0,\\lambda_{\\mathrm{c}}/2)$, we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for $\\lambda \\geq \\lambda_{\\mathrm{c}}/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-01T19:17:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "82B43"
    ],
    "keywords": [
      "biased random walk",
      "one-dimensional percolation model",
      "conditional bond percolation model",
      "central limit theorem fails",
      "regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}