{
  "id": "1803.00793",
  "version": "v1",
  "published": "2018-03-02T10:26:56.000Z",
  "updated": "2018-03-02T10:26:56.000Z",
  "title": "Equivalence of some subcritical properties in continuum percolation",
  "authors": [
    "Jean-Baptiste Gouéré",
    "Marie Théret"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the Boolean model on $\\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $\\P(\\ell \\ge n) \\le Ae^{-Bn}$ for all $n \\ge 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-02T10:26:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum percolation",
      "subcritical properties",
      "equivalence",
      "boolean model",
      "optimal integrability conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}