Asymptotics of the visibility function in the Boolean model

TL;DR

This paper provides precise tail probability estimates for the visibility function in a Boolean model, analyzing asymptotic behavior in multiple dimensions and establishing convergence to extreme value distributions under specific conditions.

Contribution

It offers new asymptotic estimates and convergence results for the visibility function in germ-grain models, extending understanding in stochastic geometry.

Findings

Tail probability estimates for the visibility function

Convergence to a type I extreme value distribution in specific cases

Analysis in multiple dimensions using coverage techniques

Abstract

The aim of this paper is to give a precise estimate on the tail probability of the visibility function in a germ-grain model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle in a Boolean model. We proceed in two or more dimensions using coverage techniques. Moreover, convergence results involving a type I extreme value distribution are shown in the two particular cases of small obstacles or a large obstacle-free region.