{
  "id": "0711.0307",
  "version": "v3",
  "published": "2007-11-02T13:34:45.000Z",
  "updated": "2007-11-21T19:22:01.000Z",
  "title": "Continuum percolation at and above the uniqueness treshold on homogeneous spaces",
  "authors": [
    "Johan Tykesson"
  ],
  "comment": "16 pages, corrections made",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the Poisson Boolean model of continuum percolation on a homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the Poisson process in the model and let $lambda_u$ be the infimum of the set of intensities that a.s. produce a unique unbounded component. We show that above $\\lambda_u$ there is a.s. a unique unbounded component. We also study what happens at $\\lambda_u$ for some spaces. In particular, if $M$ is the product of the hyperbolic disc and the real line, then at $\\lambda_u$ there is a.s. not a unique unbounded component. The results are inspired by results for Bernoulli bond percolation on graphs due to Haggstrom, Peres and Schonmann.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-11-21T19:22:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "continuum percolation",
      "uniqueness treshold",
      "unique unbounded component",
      "homogeneous spaces",
      "poisson boolean model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.0307T"
    }
  }
}