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

TL;DR

This paper develops an asymptotic theory for large-n 3D Poisson-Voronoi cells and faces, validated by extensive Monte Carlo simulations, revealing geometric scaling laws and seed distributions.

Contribution

It extends existing models to n-edged faces, predicts geometric properties at large n, and provides a comprehensive Monte Carlo dataset for validation.

Findings

The theory accurately predicts volume and surface area scaling for large n.

Monte Carlo data confirms the predicted seed distributions and geometric behaviors.

Deviations are explained by subleading correction estimates.

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/lambda (with lambda 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*10^9 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.