Asymptotics of visibility in the hyperbolic plane

At each point of a Poisson point process of intensity λ in the hyperbolic place, 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 that if λ is strictly smaller than a critical intensity λgv then Pr does not go to 0 as r→∞. The main result in this note shows that in the case λ=λgv, the probability of reaching distance larger than r decays essentially polynomial, while if λ>λgv, the decay is exponential. We also extend these results to various related models