Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell

TL;DR

This paper derives precise asymptotic estimates for the tail distribution of the maximum distance from the nucleus to a vertex in a typical Poisson-Voronoi cell, confirming a conjecture about the extremal index.

Contribution

It provides sharp asymptotics for the tail distribution of the maximal vertex distance and confirms a conjecture on the extremal index for Poisson-Voronoi cells.

Findings

Asymptotic tail distribution of the maximal distance D is characterized.

The extremal index for distances in large Voronoi cells is (2d) -1.

The explicit constant in tail estimates equals the mean volume of a certain random simplex.

Abstract

We consider the typical Poisson-Voronoi cell in the Euclidean space R d and in particular the maximal distance D from a vertex of that cell to its nucleus. We provide a sharp asymptotics for the tail distribution of D. As a byproduct, we prove that the extremal index related to the sequence of such distances for all Voronoi cells included in a large box is equal to (2d) -1 . This confirms a conjecture formulated by Chenavier and Robert. The explicit constant appearing in the estimate of the tail probability of D is proved to be the mean volume of a random simplex formed by uniformly distributed points on the unit sphere conditioned on satisfying some spatial condition.