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Limits of Poisson-Laguerre tessellations

Anna Gusakova, Mathias in Wolde-Lübke

TL;DR

This work develops a robust probabilistic framework for the limits of Poisson-Laguerre tessellations and their duals when height densities vary. By leveraging a paraboloid growth/hull representation and stabilization arguments, the authors prove two primary regimes: convergence within the Poisson-Laguerre class under local density convergence, and homogeneous limits yielding Poisson-Voronoi and Poisson-Delaunay skeleta as $h$-mass concentrates near zero. They also connect these results to typical-cell convergence, and demonstrate corollaries for β-, Gaussian-Voronoi, and related tessellations, including the canonical Poisson-Voronoi limit as a β-limit at $\beta\to-1$. The findings unify weighted Laguerre models with classical Poisson tessellations and provide tools for analyzing convergence of complex random tessellations in high dimensions.

Abstract

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(η_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $η_n$ 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 $β$-Voronoi and the $β$-Delaunay tessellations, respectively, as $β\to -1$.

Limits of Poisson-Laguerre tessellations

TL;DR

This work develops a robust probabilistic framework for the limits of Poisson-Laguerre tessellations and their duals when height densities vary. By leveraging a paraboloid growth/hull representation and stabilization arguments, the authors prove two primary regimes: convergence within the Poisson-Laguerre class under local density convergence, and homogeneous limits yielding Poisson-Voronoi and Poisson-Delaunay skeleta as -mass concentrates near zero. They also connect these results to typical-cell convergence, and demonstrate corollaries for β-, Gaussian-Voronoi, and related tessellations, including the canonical Poisson-Voronoi limit as a β-limit at . The findings unify weighted Laguerre models with classical Poisson tessellations and provide tools for analyzing convergence of complex random tessellations in high dimensions.

Abstract

For sequences of Poisson-Laguerre tessellations and their duals in , generated by Poisson point processes in , we prove limit theorems as . The intensity measure of has density of the form with respect to the Lebesgue measure, where and . Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on 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 -Voronoi and the -Delaunay tessellations, respectively, as .
Paper Structure (30 sections, 18 theorems, 234 equations, 2 figures)

This paper contains 30 sections, 18 theorems, 234 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model (Poisson-Laguerre tessellation).
  3. Convergence of random tessellations.
  4. Main results.
  5. Structure of the paper.
  6. Preliminaries
  7. Frequently used notation
  8. Random Laguerre tessellations
  9. Tessellations.
  10. Construction of Laguerre tessellations and their duals.
  11. Models of random tessellations.
  12. Limit theorems for random Laguerre tessellations
  13. Convergence of random tessellations and random closed sets
  14. Convergence in the class of Poisson-Laguerre tessellations with densities
  15. Poisson-Voronoi tessellation as weak limit of Poisson-Laguerre tessellations
  16. ...and 15 more sections

Key Result

Proposition 3.2

If $(Z_n)_{n\in\mathbb{N}}$ converges to $Z$ locally with high probability, then $(Z_n)_{n\in\mathbb{N}}$ converges to $Z$ in probability and weakly.

Figures (2)

  • Figure 1: The paraboloid growth (black) and hull processes (red) in $\mathbb{R}^2$.
  • Figure 2: The downward parabola $(\Pi^-(\mathbf{v},\mathbf{h}))^{\downarrow}$ and the stripe $(B(\mathbf{v})\times [0,9r_n^2/4])$ in $\mathbb{R}^2$.

Theorems & Definitions (35)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Remark 3.7
  • Theorem 3.8
  • Remark 3.9
  • ...and 25 more