Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation

Abstract

This work considers the 3D Poisson-Voronoi tessellation. It investigates the joint probability distribution for an arbitrarily selected cell face to be n-edged and for the distance between the seeds of the two adjacent cells to be equal to 2L. For this quantity an exact expression is derived, valid in the limit with n1/6L fixed. The leading order correction term is determined. Good agreement with earlier Monte Carlo data is obtained. The cell face is shown to be surrounded by a three-dimensional domain that is empty of seeds and is the union of n balls; it is pumpkin-shaped and analogous to the flower of the 2D Voronoi cell. For this domain tends towards a torus of equal major and minor radii. The radii scale as n1/3, in agreement with earlier heuristic work. A detailed understanding is achieved of several other statistical properties of the n-edged cell face.