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Asymptotics of visibility in the hyperbolic plane. (English) Zbl 1276.60012
Summary: At each point of a Poisson point process of intensity \(\lambda \) in the hyperbolic plane, center a ball of bounded random radius. Consider the probability \(P_{r}\) that, from a fixed point, there is some direction in which one can reach distance \(r\) without hitting any ball. It is known [Benjamini et al., “Visibility to infinity in the hyperbolic plane, despite obstacles”, ALEA, Lat. Am. J. Probab. Math. Stat. 6, 323–342 (2009)] that, if \(\lambda \) is strictly smaller than a critical intensity \(\lambda _{ gv}\), then \(P_{r}\) does not go to zero as \(r \rightarrow \infty \). The main result in this note shows that, in the case \(\lambda =\lambda _{gv}\), the probability of reaching a distance larger than \(r\) decays essentially polynomially, while, if \(\lambda >\lambda _{gv}\), the decay is exponential. We also extend these results to various related models, and we finally obtain asymptotic results in several situations.

MSC:
60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Software:
GVF
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