Asymptotics of visibility in the hyperbolic plane

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 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 (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λ _gv thenP_r does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λ_gv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ>λ_gv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.