The volume of simplices in high-dimensional Poisson-Delaunay tessellations

TL;DR

This paper investigates the volume distribution of weighted simplices in high-dimensional Poisson-Delaunay tessellations, establishing a central limit theorem and analyzing deviations, with results applicable to various weight parameters and dimension-dependent cases.

Contribution

It provides the first sharp bounds on cumulants and proves a central limit theorem for the logarithmic volume of weighted simplices in high dimensions.

Findings

Logarithmic volume satisfies a central limit theorem as dimension grows.

Provides rates of convergence and concentration inequalities.

Analyzes large deviations and mod-$ ext{phi}$ convergence for fixed weights.

Abstract

Typical weighted random simplices $Z_{\mu}$, $\mu\in(-2,\infty)$, in a Poisson-Delaunay tessellation in $\mathbb{R}^n$ are considered, where the weight is given by the $(\mu+1)$st power of the volume. As special cases this includes the typical ($\mu=-1$) and the usual volume-weighted ($\mu=0$) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_{\mu}$ satisfies a central limit theorem in high dimensions, that is, as $n\to\infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight $\mu=\mu(n)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $\mu$ also mod-$\phi$ convergence and the large deviations behaviour of the logarithmic volume of $Z_{\mu}$ are investigated.