{
  "id": "math/0503544",
  "version": "v1",
  "published": "2005-03-24T14:12:18.000Z",
  "updated": "2005-03-24T14:12:18.000Z",
  "title": "Continuum percolation with steps in an annulus",
  "authors": [
    "Paul Balister",
    "Bela Bollobas",
    "Mark Walters"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/105051604000000891 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2004, Vol. 14, No. 4, 1869-1879",
  "doi": "10.1214/105051604000000891",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let A be the annulus in R^2 centered at the origin with inner and outer radii r(1-\\epsilon) and r, respectively. Place points {x_i} in R^2 according to a Poisson process with intensity 1 and let G_A be the random graph with vertex set {x_i} and edges x_ix_j whenever x_i-x_j\\in A. We show that if the area of A is large, then G_A almost surely has an infinite component. Moreover, if we fix \\epsilon, increase r and let n_c=n_c(\\epsilon) be the area of A when this infinite component appears, then n_c\\to1 as \\epsilon \\to 0. This is in contrast to the case of a ``square'' annulus where we show that n_c is bounded away from 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-24T14:12:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "continuum percolation",
      "infinite component appears",
      "poisson process",
      "random graph",
      "place points"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3544B"
    }
  }
}