Asymptotics of visibility in the hyperbolic plane

Details Asymptotics of visibility in the hyperbolic plane Asymptotics of visibility in the hyperbolic plane

2010-12-23

arxiv.org/abs/1012.5220v2

Mathematics

Probability

17 pages, preliminary version. Version 2: minor corrections and a minor structural change

Scientific paper

At each point of a Poisson point process of intensity $\lambda$ in the hyperbolic place, 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 \cite{BJST} that if $\lambda$ is strictly smaller than a critical intensity $\lambda_{gv}$ then $P_r$ does not go to $0$ as $r\to \infty$. The main result in this note shows that in the case $\lambda=\lambda_{gv}$, the probability of reaching distance larger than $r$ decays essentially polynomial, while if $\lambda>\lambda_{gv}$, the decay is exponential. We also extend these results to various related models.

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