The $\beta$-Delaunay tessellation II: The Gaussian limit tessellation

TL;DR

This paper investigates the convergence of $eta$-Delaunay tessellations to a Gaussian limit tessellation in $R^{d-1}$, characterizing its structure and geometric properties, and linking it to Gaussian random simplices.

Contribution

It introduces the Gaussian-Delaunay tessellation as a new stationary random tessellation derived from the limit of $eta$-Delaunay tessellations, with explicit distributional and geometric analysis.

Findings

Explicit distribution of volume-weighted typical cells

Calculation of volume moments and angle sums

Determination of cell intensities

Abstract

We study the weak convergence of $\beta$- and $\beta'$-Delaunay tessellations in $\mathbb{R}^{d-1}$ that were introduced in part I of this paper, as $\beta\to\infty$. The limiting stationary simplicial random tessellation, which is called the Gaussian-Delaunay tessellation, is characterized in terms of a space-time paraboloid hull process in $\mathbb{R}^{d-1}\times\mathbb{R}$. The latter object has previously appeared in the analysis of the number of shocks in the solution of the inviscid Burgers' equation and the description of the local asymptotic geometry of Gaussian random polytopes. In this paper it is used to define a new stationary random simplicial tessellation in $\mathbb{R}^{d-1}$. As for the $\beta$- and $\beta'$-Delaunay tessellation, the distribution of volume-power weighted typical cells in the Gaussian-Delaunay tessellation is explicitly identified, establishing thereby a new bridge to Gaussian random simplices. Also major geometric characteristics of these cells such as volume moments, expected angle sums and also the cell intensities of the Gaussian-Delaunay tessellation are investigated.