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Asymptotics of Visibility in the Hyperbolic Plane

Published online by Cambridge University Press:  22 February 2016

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.

Type
Stochastic Geometry and Statistical Applications
Copyright
© Applied Probability Trust 

Footnotes

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

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