{
  "id": "math/0304418",
  "version": "v4",
  "published": "2003-04-26T01:03:12.000Z",
  "updated": "2005-04-06T06:30:29.000Z",
  "title": "On the scaling of the chemical distance in long-range percolation models",
  "authors": [
    "Marek Biskup"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117904000000577 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2004, Vol. 32, No. 4, 2938-2977",
  "doi": "10.1214/009117904000000577",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the (unoriented) long-range percolation on Z^d in dimensions d\\ge1, where distinct sites x,y\\in Z^d get connected with probability p_{xy}\\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\\to\\infty, where s>0 and |\\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\\infty}. Our main result is that, for s\\in(d,2d), D(x,y)=(\\log|x-y|)^{\\Delta+o(1)},\\qquad x,y\\in C_{\\infty}, |x-y|\\to\\infty, where \\Delta^{-1} is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x-y|\\to\\infty. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of ``small-world'' phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2005-04-06T06:30:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "82B28"
    ],
    "keywords": [
      "long-range percolation models",
      "chemical distance",
      "probability",
      "resulting random graph contains",
      "finite box contains"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......4418B"
    }
  }
}