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Degrees in the $β$- and $β'$-Delaunay graphs

Gilles Bonnet, Joseph Gordon

TL;DR

This work analyzes the typical cells and degree statistics of the $\beta$- and $\beta'$-Delaunay/Voronoi tessellations arising from Poisson-Laguerre constructions in $\mathbb{R}^d$. Central to the approach is the Complementary Theorem, which decomposes the typical cell into a $\Phi$-content (gamma-distributed given facet count) and a shape component that is independent of the $\Phi$-content; this yields integral representations and enables sharp bounds on facet and degree distributions. The paper proves a lower bound for $\mathbb{P}(Z^{(\prime)}\in\mathcal{P}_n)$ that is exponential in $n$, and an upper bound for the $\beta$-case with a super-exponential decay, together with a concentration result for the maximal degree in growing windows; for $d=2$ this concentration reduces to two attainable values. The results illuminate a marked contrast between the $\beta$- and $\beta'$-models (exponential vs. super-exponential tails) and advance understanding of maximal degrees in random spatial graphs driven by tessellations, with potential implications for extremal geometry and stochastic geometry of Laguerre diagrams.

Abstract

We investigate the typical cells $\widehat{Z}$ and $\widehat{Z}^\prime$ of $β$- and $β'$-Voronoi tessellations in $\mathbb{R}^d$, establishing a Complementary Theorem which entails: 1) a gamma distribution of the $Φ$-content (a suitable homogeneous functional) of the typical cell with $n$-facets; 2) the independence of this $Φ$-content with the shape of the cell; 3) a practical integral representation of the distribution of $Z^{(\prime)}$. We exploit the latter to derive bounds on the distribution of the facet numbers. Using duality, we get bounds on the typical degree distributions of $β$- and $β'$-Delaunay triangulations. For $β'$-Delaunay, the resulting exponential lower bound seems to be the first of its kind for random spatial graphs arising as the skeletons of random tessellations. For $β$-Delaunay, matching super-exponential bounds allow us to show concentration of the maximal degree in a growing window to only a finite number of deterministic values (in particular, only two values for $d=2$).

Degrees in the $β$- and $β'$-Delaunay graphs

TL;DR

This work analyzes the typical cells and degree statistics of the - and -Delaunay/Voronoi tessellations arising from Poisson-Laguerre constructions in . Central to the approach is the Complementary Theorem, which decomposes the typical cell into a -content (gamma-distributed given facet count) and a shape component that is independent of the -content; this yields integral representations and enables sharp bounds on facet and degree distributions. The paper proves a lower bound for that is exponential in , and an upper bound for the -case with a super-exponential decay, together with a concentration result for the maximal degree in growing windows; for this concentration reduces to two attainable values. The results illuminate a marked contrast between the - and -models (exponential vs. super-exponential tails) and advance understanding of maximal degrees in random spatial graphs driven by tessellations, with potential implications for extremal geometry and stochastic geometry of Laguerre diagrams.

Abstract

We investigate the typical cells and of - and -Voronoi tessellations in , establishing a Complementary Theorem which entails: 1) a gamma distribution of the -content (a suitable homogeneous functional) of the typical cell with -facets; 2) the independence of this -content with the shape of the cell; 3) a practical integral representation of the distribution of . We exploit the latter to derive bounds on the distribution of the facet numbers. Using duality, we get bounds on the typical degree distributions of - and -Delaunay triangulations. For -Delaunay, the resulting exponential lower bound seems to be the first of its kind for random spatial graphs arising as the skeletons of random tessellations. For -Delaunay, matching super-exponential bounds allow us to show concentration of the maximal degree in a growing window to only a finite number of deterministic values (in particular, only two values for ).

Paper Structure

This paper contains 23 sections, 34 theorems, 187 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Basics
  4. Half-spaces, polytopes, convex bodies
  5. Tessellations
  6. Paraboloids
  7. Functions and constants
  8. The models
  9. Deterministic constructions
  10. Random tessellations
  11. Complementary theorem
  12. Typical cell
  13. Arcs and measures on the space of half-spaces
  14. Flower and $\Phi$-content
  15. Typical cell with $n$ facets
  16. ...and 8 more sections

Key Result

Proposition 2.1

Let $x > 0$ and $a\in(0,\max(x,1))$. Then

Figures (7)

  • Figure 1: First row: $\beta$-Delaunay and $\beta'$-Delaunay triangulations with $\beta=3$. Second row: corresponding dual Voronoi tessellations.
  • Figure 2: Paraboloids in $\mathbb{R}^1\times\mathbb{R}$.
  • Figure 3: A Weighted Delaunay triangulation on $\mathbb{R}$. Illustration courtesy of Anna Gusakova.
  • Figure 4: A Voronoi flower of a point in $\mathbb{R}^2$, view from the side and from above
  • Figure 5: The curve $\operatorname{arc}(x,H^-)$ for some $x=(\mathbf{o},h)$, $h\in\mathbb{R}$, and $H^-=H^-(u,t)\in\mathcal{H}^-$. Geometrically it can be constructed as follows. We (1) take the downwards parabola with apex above $tu$ containing $x$; (2) divide it into two arcs at $x$; (3) take the arc that has an intersection of infinite length with $H^-\times\mathbb{R}$.
  • ...and 2 more figures

Theorems & Definitions (76)

  • Proposition 2.1: gautschi_1959gabcke_1979
  • Remark 2.2
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4: gusakova_poisson-laguerre_2024 Correctness of the model
  • Lemma 3.5: Properties of intensity measures
  • proof
  • Remark 3.6
  • Corollary 3.7
  • ...and 66 more