Asymptotics of visibility in the hyperbolic plane

Johan Tykesson (1) , Pierre Calka (2)

1 Department of Computer Science and Applied Mathematics [Rehovot]
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

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 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 and we finally obtain asymptotic results in several situations.

Domaines: Probabilités [math.PR]

Johan Tykesson, Pierre Calka. Asymptotics of visibility in the hyperbolic plane. 2011. ⟨hal-00551930⟩