Asymptotics of visibility in the hyperbolic plane

TL;DR

This paper investigates the asymptotic behavior of the probability of reaching large distances without hitting obstacles in a hyperbolic plane with a Poisson point process, revealing different decay rates at and above a critical intensity.

Contribution

It establishes the decay rates of the reachability probability at the critical and supercritical intensities in hyperbolic space, extending previous results.

Findings

At critical intensity, reachability probability decays polynomially.

Above critical intensity, decay is exponential.

Results extend to related geometric models.

Abstract

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 $r\to \infty$. 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.