The maximal degree in a Poisson-Delaunay graph

We investigate the maximal degree in a Poisson-Delaunay graph in R-d, d >= 2, over all nodes in the window 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 >= 3.