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June 2013 Asymptotics of visibility in the hyperbolic plane
Johan Tykesson, Pierre Calka
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Adv. in Appl. Probab. 45(2): 332-350 (June 2013). DOI: 10.1239/aap/1370870121

Abstract

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 λ gv 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.

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Johan Tykesson. Pierre Calka. "Asymptotics of visibility in the hyperbolic plane." Adv. in Appl. Probab. 45 (2) 332 - 350, June 2013. https://doi.org/10.1239/aap/1370870121

Information

Published: June 2013
First available in Project Euclid: 10 June 2013

zbMATH: 1276.60012
MathSciNet: MR3102454
Digital Object Identifier: 10.1239/aap/1370870121

Subjects:
Primary: 60D05
Secondary: 60G55

Rights: Copyright © 2013 Applied Probability Trust

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Vol.45 • No. 2 • June 2013
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