{
  "id": "math/0411560",
  "version": "v3",
  "published": "2004-11-24T17:58:00.000Z",
  "updated": "2006-06-22T12:53:49.000Z",
  "title": "The time constant and critical probabilities in percolation models",
  "authors": [
    "Leandro P. R. Pimentel"
  ],
  "comment": "8 pages, 2 figures",
  "journal": "Elect. Comm. in Probab. 11 (2006), 160--167",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a first-passage percolation model on a Delaunay triangulation of the plane. In this model each edge is independently equipped with a nonnegative random variable, with distribution function F, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman have shown that, under a suitable moment condition on F, the minimum time taken to reach a point x from the origin 0 is asymptotically \\mu(F)|x|, where \\mu(F) is a nonnegative finite constant. However the exact value of the time constant \\mu(F) still a fundamental problem in percolation theory. Here we prove that if F(0)<1-p_c^* then \\mu(F)>0, where p_c^* is a critical probability for bond percolation on the dual graph (Voronoi tessellation).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-06-22T12:53:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82D30"
    ],
    "keywords": [
      "time constant",
      "critical probability",
      "first-passage percolation model",
      "minimum time taken",
      "delaunay triangulation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11560P"
    }
  }
}