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 $rto infty$. 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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