{
  "id": "1605.07598",
  "version": "v1",
  "published": "2016-05-24T19:47:43.000Z",
  "updated": "2016-05-24T19:47:43.000Z",
  "title": "Ellipses Percolation",
  "authors": [
    "Augusto Teixeira",
    "Daniel Ungaretti"
  ],
  "comment": "29 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We define a continuum percolation model that provides a collection of random ellipses on the plane and study the behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter $\\alpha > 0$ associated with the tail decay of the major axis distribution; we only consider distributions $\\rho$ satisfying $\\rho[r, \\infty) \\asymp r^{-\\alpha}$. We prove that this model presents a double phase transition in $\\alpha$. For $\\alpha \\in (0,1]$ the plane is completely covered by the ellipses, almost surely. For $\\alpha \\in (1,2)$ the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For $\\alpha \\in (2, \\infty)$ the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter $\\alpha = 2$ that there is a non-degenerate interval of density for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one, a rather unusual phenomenon. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on $\\mathbb{Z}^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-24T19:47:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "60G55"
    ],
    "keywords": [
      "ellipses percolation",
      "vacant set percolate",
      "continuum percolation model",
      "poisson cylinder model",
      "major axis distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}