{
  "id": "0807.3308",
  "version": "v1",
  "published": "2008-07-21T16:39:29.000Z",
  "updated": "2008-07-21T16:39:29.000Z",
  "title": "Visibility to infinity in the hyperbolic plane, despite obstacles",
  "authors": [
    "Itai Benjamini",
    "Johan Jonasson",
    "Oded Schramm",
    "Johan Tykesson"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Suppose that $Z$ is a random closed subset of the hyperbolic plane $\\H^2$, whose law is invariant under isometries of $\\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in $\\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays fast enough, then there are almost surely no lines in $Z$. We also show that if the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$. In the case of the Poisson Boolean model with balls of fixed radius $R$ we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the Grassmannian of lines in $\\H^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-21T16:39:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B43",
      "82B27",
      "82B21"
    ],
    "keywords": [
      "hyperbolic plane",
      "despite obstacles",
      "poisson boolean model",
      "visibility",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.3308B"
    }
  }
}