{
  "id": "1012.3386",
  "version": "v1",
  "published": "2010-12-15T16:57:07.000Z",
  "updated": "2010-12-15T16:57:07.000Z",
  "title": "On the speed of biased random walk in translation invariant percolation",
  "authors": [
    "Maria Deijfen",
    "Olle Häggström"
  ],
  "journal": "ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010) 19-40",
  "categories": [
    "math.PR"
  ],
  "abstract": "For biased random walk on the infinite cluster in supercritical i.i.d.\\ percolation on $\\Z^2$, where the bias of the walk is quantified by a parameter $\\beta>1$, it has been conjectured (and partly proved) that there exists a critical value $\\beta_c>1$ such that the walk has positive speed when $\\beta<\\beta_c$ and speed zero when $\\beta>\\beta_c$. In this paper, biased random walk on the infinite cluster of a certain translation invariant percolation process on $\\Z^2$ is considered. The example is shown to exhibit the opposite behavior to what is expected for i.i.d.\\ percolation, in the sense that it has a critical value $\\beta_c$ such that, for $\\beta<\\beta_c$, the random walk has speed zero, while, for $\\beta>\\beta_c$, the speed is positive. Hence the monotonicity in $\\beta$ that is part of the conjecture for i.i.d.\\ percolation cannot be extended to general translation invariant percolation processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-15T16:57:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "biased random walk",
      "general translation invariant percolation processes",
      "infinite cluster",
      "speed zero",
      "critical value"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3386D"
    }
  }
}