Random cover times using the Poisson cylinder process

TL;DR

This paper studies the distribution of the time it takes for a Poisson process of cylinders to cover a set in Euclidean space, revealing asymptotic behaviors and convergence types depending on the set's structure.

Contribution

It introduces a model of random cover times using a Poisson cylinder process and characterizes the asymptotic distribution of cover times for different set geometries.

Findings

Convergence to Gumbel distribution for discrete, well-separated sets.

Determines the rate of convergence for sets with positive box dimension.

Provides asymptotic results for the cover time as the set size increases.

Abstract

In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set $A \subset \mathbb{R}^d.$ This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are "raining from the sky" at unit rate. Our main results concerns the asymptotic of this cover time as the set $A$ grows. If the set $A$ is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead $A$ has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence.