Limits of Poisson-Laguerre tessellations

TL;DR

This paper establishes limit theorems for sequences of Poisson-Laguerre tessellations and their duals in Euclidean space, identifying conditions under which they converge to classical or other Poisson tessellations, and explores convergence of typical cells.

Contribution

It provides verifiable conditions on the intensity functions that guarantee convergence of Poisson-Laguerre tessellations to classical or alternative Poisson tessellations, including the Poisson-Voronoi and Delaunay types.

Findings

Convergence conditions for tessellations to classical Poisson models.

Identification of limit behaviors for typical cells.

Poisson-Voronoi and Delaunay tessellations as limits of $eta$-models.

Abstract

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(\eta_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $\eta_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n)_{n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $\beta$-Voronoi and the $\beta$-Delaunay tessellations, respectively, as $\beta\to -1$.