The maximal degree in a Poisson-Delaunay graph

Abstract

We investigate the maximal degree in a Poisson-Delaunay graph in ${\mathbf{R}}^d$, $d\ge 2$, 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 $d\ge 3$