On the maximum visibility in a ball through the vacant set of Poissonian obstacles

We study the maximum visibility in a ball inside the vacant set of three obstacle models in ${\mathbb{R}}^d$ with slow decay of spatial correlations and disparate obstacle geometries: Poisson Boolean models with general i.i.d. radii distributions, Poisson cylinders and Brownian interlacements. Let $M_r$ be the maximum distance between points $x$ and $y$ in the ball $B(r)$ such that $x$ is visible from $y$. We prove that $M_r$ divided by $q_r$ converges in probability to an explicit model dependent constant, where $q_r=\log r$, except for the Brownian interlacements in dimension $d=3$, where $q_r=\log r\log \log r$.