Asymptotics of visibility in the hyperbolic plane

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 $r\to \mathrm{\infty }$. 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.