New Monte Carlo method for planar Poisson-Voronoi cells

TL;DR

This paper introduces a new Monte Carlo algorithm to accurately evaluate properties of planar Poisson-Voronoi cells with up to 1600 sides, analyzing their asymptotic behavior and rare configurations.

Contribution

A novel Monte Carlo method based on theoretical insights for precise computation of Poisson-Voronoi cell properties up to 1600 sides.

Findings

Accurate p_n values for 3 ≤ n ≤ 1600 with 4-6 significant digits

Asymptotic power series describing area and perimeter statistics

Visualization of rare, many-sided cell configurations

Abstract

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have between four and six significant digits. Accurate n dependent averages, second moments, and variances are obtained for the cell area and the cell perimeter. The numerical large n behavior of these quantities is analyzed in terms of asymptotic power series in 1/n. Snapshots are shown of typical occurrences of extremely rare events implicating cells of up to n=1600 sides embedded in an ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.