{
  "id": "1012.5220",
  "version": "v2",
  "published": "2010-12-23T14:53:16.000Z",
  "updated": "2011-01-14T13:39:36.000Z",
  "title": "Asymptotics of visibility in the hyperbolic plane",
  "authors": [
    "Pierre Calka",
    "Johan Tykesson"
  ],
  "comment": "17 pages, preliminary version. Version 2: minor corrections and a minor structural change",
  "categories": [
    "math.PR"
  ],
  "abstract": "At each point of a Poisson point process of intensity $\\lambda$ in the hyperbolic place, center a ball of bounded random radius. Consider the probability $P_r$ that from a fixed point, there is some direction in which one can reach distance $r$ without hitting any ball. It is known \\cite{BJST} that if $\\lambda$ is strictly smaller than a critical intensity $\\lambda_{gv}$ then $P_r$ does not go to $0$ as $r\\to \\infty$. The main result in this note shows that in the case $\\lambda=\\lambda_{gv}$, the probability of reaching distance larger than $r$ decays essentially polynomial, while if $\\lambda>\\lambda_{gv}$, the decay is exponential. We also extend these results to various related models.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-01-14T13:39:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B43",
      "82B27",
      "82B21"
    ],
    "keywords": [
      "hyperbolic plane",
      "asymptotics",
      "visibility",
      "poisson point process",
      "hyperbolic place"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.5220C"
    }
  }
}