The maximal degree in a Poisson–Delaunay graph

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. A weaker version of this result is discussed when $d\geq 3$.