## Hausdorff dimension of visible sets for well-behaved continuum percolation in the hyperbolic plane

Autor(en): | Thaele, Christoph |

Stichwörter: | Boolean model; continuum percolation; fractal geometry; Hausdorff-dimension; hyperbolic geometry; Mathematics; Statistics & Probability |

Erscheinungsdatum: | 2014 |

Herausgeber: | BRAZILIAN STATISTICAL ASSOCIATION |

Journal: | BRAZILIAN JOURNAL OF PROBABILITY AND STATISTICS |

Volumen: | 28 |

Ausgabe: | 1 |

Startseite: | 73 |

Seitenende: | 82 |

Zusammenfassung: | Let Z be a so-called well-behaved percolation, that is, a certain random closed set in the hyperbolic plane, whose law is invariant under all isometries; for example, the covered region in a Poisson Boolean model. In terms of the alpha-value of Z, the Hausdorff-dimension of the set of directions is determined in which visibility from a fixed point to the ideal boundary of the hyperbolic plane is possible within Z. Moreover, the Hausdorff-dimension of the set of (hyperbolic) lines through a fixed point contained in Z is calculated. Thereby several conjectures raised by Benjamini, Jonasson, Schramm and Tykesson are confirmed. |

ISSN: | 01030752 |

DOI: | 10.1214/12-BJPS194 |

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