The maximal degree in a Poisson-Delaunay graph

TL;DR

This paper analyzes the maximum degree in Poisson-Delaunay graphs in multi-dimensional space, providing exact order results and concentration properties, especially in two dimensions, as the observation window grows infinitely large.

Contribution

It determines the exact order of the maximum degree in Poisson-Delaunay graphs across all dimensions and shows high-probability concentration in two dimensions.

Findings

Maximum degree order is explicitly characterized for all dimensions.

In 2D, the maximum degree concentrates on two integers with high probability.

Results extend to higher dimensions with discussion on their implications.

Abstract

We investigate the maximal degree in a Poisson-Delaunay graph in $\mathbf{R}^d$, $d\geq 2$, over all nodes in the window $\mathbf{W}_\rho:= \rho^{1/d}[0,1]^d$ as $\rho$ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting $d=2$, we show that this quantity is concentrated on two consecutive integers with high probability. An extension of this result is discussed when $d\geq 3$