Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

Abstract

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, . . . . First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ −1 (1−e −λu ). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

Thin-shell concentration for zero cells of stationary Poisson mosaics

“…This implies that the random vector of interest will be radially symmetric. In the Poisson-Voronoi case, we show that this random vector is also log-concave, and thus satisfies the thin-shell estimate (1). We also prove strong deviation estimates by direct computation.…”

“…by the thin-shell estimate (1). We can also prove strong concentration inequalities by direct computation.…”

“…For these applications, it is important to understand the asymptotic geometric properties of the convex polytopes induced by random tessellations, in order to decode and reconstruct high dimensional signals with small error. Some recent work has considered high dimensional Poisson mosaics in particular, focusing on the volume and shape of the zero cell and typical cell as dimension n tends to infinity [1,9,10]. For example, in [1], it is proved that the volume of the intersection of the typical cell of a Poisson-Voronoi mosaic with intensity λ and a co-centered ball of volume u tends to λ −1 (1 − e −λu ) as the space dimension tends to infinity.…”