Extreme Distance Distributions of Poisson Voronoi Cells

Abstract

Poisson point processes provide a versatile framework for modeling the distributions of random points in space. When the space is partitioned into cells, each associated with a single generating point from the Poisson process, there appears a geometric structure known as Poisson Voronoi tessellation. These tessellations find applications in various fields such as biology, material science, and communications, where the statistical properties of the Voronoi cells reveal patterns and structures that hold key insights into the underlying processes generating the observed phenomena. In this paper, we investigate a distance measure of Poisson Voronoi tessellations that is emerging in the literature, yet for which its statistical and geometrical properties remain explored only in the asymptotic case when the density of seed points approaches infinity. Our work, specifically focused on homogeneous Poisson point processes, characterizes the cumulative distribution functions governing the smallest and largest distances between the points generating the Voronoi regions and their respective vertices for an arbitrary density of points in $\mathbb{R}^2$. For that, we conduct a Monte-Carlo type simulation with $10^8$ Voronoi cells and fit the resulting empirical cumulative distribution functions to the Generalized Gamma, Gamma, Log-normal, Rayleigh, and Weibull distributions. Our analysis compares these fits in terms of root mean-squared error and maximum absolute variation, revealing the Generalized Gamma distribution as the best-fit model for characterizing these distances in homogeneous Poisson Voronoi tessellations. Furthermore, we provide estimates for the maximum likelihood and the $95$\% confidence interval of the parameters of the Generalized Gamma distribution along with the algorithm implemented to calculate the maximum and minimum distances.