Thin-shell concentration for zero cells of stationary Poisson mosaics

TL;DR

This paper investigates how the norm of a uniformly sampled point in the zero cell of stationary Poisson mosaics concentrates around a predictable value as the dimension grows large, revealing different behaviors for Voronoi and hyperplane mosaics.

Contribution

It provides the first detailed analysis of the high-dimensional concentration phenomena for zero cells in stationary Poisson mosaics, including explicit asymptotic behaviors and convergence rates.

Findings

For Poisson-Voronoi mosaics, the normalized radius approaches a specific constant as dimension increases.

For Poisson hyperplane mosaics, the radius concentrates within a range with high probability.

The convergence rates of the concentration phenomena are explicitly derived.

Abstract

We study the concentration of the norm of a random vector $Y$ uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in $\mathbb{R}^n$ for large dimensions $n$. For a stationary and isotropic Poisson-Voronoi mosaic, $Y$ has a radial and log-concave distribution, implying that ${|Y|}/{\mathbb{E}(|Y|^2)^{\frac{1}{2}}}$ approaches one for large $n$. Assuming the cell intensity of the random mosaic scales like $e^{n \rho_n}$, where $\lim_{n \to \infty} \rho_n = \rho$, $|Y|$ is on the order of $\sqrt{n}$ for large $n$. For the Poisson-Voronoi mosaic, we show that $|Y|/\sqrt{n}$ concentrates to $e^{-\rho}(2\pi e)^{-\frac{1}{2}}$ as $n$ increases, and for a stationary and isotropic Poisson hyperplane mosaic, we show there is a range $(R_{\ell}, R_u)$ such that ${|Y|}/{\sqrt{n}}$ will be within this range with high probability for large $n$. The rates of convergence are also computed in both cases.