The maximal degree in a Poisson-Delaunay graph

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 $d\geq 3$