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

TL;DR

This paper derives an exact asymptotic expression for the joint probability distribution of the number of edges and the distance between seed points in a 3D Poisson-Voronoi tessellation, providing new insights into large-edge cell face statistics.

Contribution

It provides the first exact asymptotic formula for the joint distribution of n-edged faces and seed distances in 3D Poisson-Voronoi tessellations, including correction terms.

Findings

Derived an exact asymptotic expression for pi_n(L) as n→∞.

Confirmed the n^{1/3} scaling of the radii of the surrounding domain.

Achieved detailed understanding of statistical properties of n-edged cell faces.

Abstract

We consider the 3D Poisson-Voronoi tessellation. We investigate the joint probability distribution pi_n(L) for an arbitrarily selected cell face to be n-edged and for the distance between the seeds of its adjacent cells to be equal to 2L. We derive an exact expression for this quantity, valid in the limit n->infty with n^{1/6}L fixed. The leading order correction term is determined. Good agreement with earlier Monte Carlo data is obtained. The cell face is surrounded by a three-dimensional excluded domain that is the union of n balls; it is pumpkin-shaped and analogous to the flower of the 2D Voronoi cell. For n->infty this domain tends towards a torus of equal major and minor radii. The radii scale as n^{1/3}, in agreement with earlier heuristic work. We achieve a detailed understanding of several other statistical properties of the n-edged cell face.