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

Yingxin Mu; Artem Sapozhnikov

paper

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

arXiv:2609.11690

Abstract

We study the maximum visibility in a ball inside the vacant set of three obstacle models in $R^{d}$ 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 $q_{r} = lo g r$ , except for the Brownian interlacements in dimension $d = 3$ , where $q_{r} = lo g r lo g lo g r$ .

arXiv comment7 pages