Degrees in the $β$- and $β'$-Delaunay graphs

Abstract

We investigate the typical cells $\widehat{Z}$ and $\widehat{Z}^\prime$ of $β$- and $β'$-Voronoi tessellations in $\mathbb{R}^d$, establishing a Complementary Theorem which entails: 1) a gamma distribution of the $Φ$-content (a suitable homogeneous functional) of the typical cell with $n$-facets; 2) the independence of this $Φ$-content with the shape of the cell; 3) a practical integral representation of the distribution of $Z^{(\prime)}$. We exploit the latter to derive bounds on the distribution of the facet numbers. Using duality, we get bounds on the typical degree distributions of $β$- and $β'$-Delaunay triangulations. For $β'$-Delaunay, the resulting exponential lower bound seems to be the first of its kind for random spatial graphs arising as the skeletons of random tessellations. For $β$-Delaunay, matching super-exponential bounds allow us to show concentration of the maximal degree in a growing window to only a finite number of deterministic values (in particular, only two values for $d=2$).