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Asymptotics of Visibility in the Hyperbolic Plane
Published online by Cambridge University Press: 22 February 2016
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 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.
MSC classification
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- Stochastic Geometry and Statistical Applications
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- Copyright
- © Applied Probability Trust
Footnotes
Research supported by a postdoctoral grant from the Swedish Research Council.
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