Many-faced cells and many-edged faces in 3D Poisson–Voronoi tessellations

Abstract

Motivated by recent new Monte Carlo data we investigate a heuristic asymptotic theory that applies to n-faced 3D Poisson–Voronoi cells in the limit of large n. We show how this theory may be extended to n-edged cell faces. It predicts the leading order large-n behavior of the average volume and surface area of the n-faced cell, and of the average area and perimeter of the n-edged face. Such a face is shown to be surrounded by a toroidal region of volume n/λ (with λ the seed density) that is void of seeds. Two neighboring cells sharing an n-edged face are found to have their seeds at a typical distance that scales as n−1/6 and whose probability law we determine. We present a new data set of 4 × 109 Monte Carlo generated 3D Poisson–Voronoi cells, larger than any before. Full compatibility is found between the Monte Carlo data and the theory. Deviations from the asymptotic predictions are explained in terms of subleading corrections whose powers in n we estimate from the data.