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Asymptotics of Visibility in the Hyperbolic Plane

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Johan Tykesson  and
Pierre Calka
Johan Tykesson*
Affiliation:
Weizmann Institute of Science, Rehovot
Pierre Calka*
Affiliation:
Université de Rouen
*
∗ Current address: Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden. Email address: johan.tykesson@gmail.com
∗∗ Postal address: Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP 12 Technopôle du Madrillet, F76801 Saint-Etienne-du-Rouvray, France. Email address: pierre.calka@univ-rouen.fr
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Abstract

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At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability Pr that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λgv thenPr does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λgv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ>λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.

Keywords

Boolean modelhyperbolic geometryvisibilityPoincaré discPoisson line process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 2 , June 2013 , pp. 332 - 350
Copyright
© Applied Probability Trust 

Footnotes

Research supported by a postdoctoral grant from the Swedish Research Council.

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