On the shape of the typical Poisson-Voronoi cell in high dimensions

TL;DR

This paper investigates the geometric properties of the typical cell in high-dimensional Poisson-Voronoi tessellations, revealing convergence behaviors and bounds for various cell metrics as the dimension grows.

Contribution

It provides new asymptotic results on the shape and face structure of the typical Poisson-Voronoi cell in high dimensions, including convergence of key geometric measures.

Findings

Inradius, outradius, diameter, and mean width converge to specific constants as dimension increases.

The width of the typical cell is bounded within explicit bounds with high probability.

The number of faces of certain dimensions grows exponentially with the dimension.

Abstract

We study the typical cell of the Poisson-Voronoi tessellation. We show that when divided by the $d$-th root of the intensity parameter $\lambda$ of the Poisson process times the volume of the unit ball, the inradius, outradius, diameter and mean width of the typical cell converge in probability to the constants $1/2, 1, 2, 2$ respectively, as the dimension $d\to\infty$. We also show that the width of the typical cell, when rescaled in the same way, is bounded between $2\sqrt{5}/(2+\sqrt{5})-o_d(1)$ and $3/2+o_d(1)$, with probability $1-o_d(1)$. These results in particular imply that, with probability $1-o_d(1)$, the Hausdorff distance between the typical cell and any ball is at least of the order of the diameter of the typical cell. In addition, we show that for all $k$ with $d-k\to\infty$, with probability $1-o_d(1)$, all faces of dimension $k$ have a diameter that is of a much smaller order than the diameter, inradius, etc., of the full typical cell. The same is true for ''almost all'' faces of dimension $d-k$ with $k$ fixed. And, we show that the number of such faces is $\left( (k+1)^{(k+1)/2} / k^{k/2} \pm o_d(1) \right)^d$ with probability $1-o_d(1)$.