Heuristic theory for many-faced d-dimensional Poisson-Voronoi cells

TL;DR

This paper extends heuristic methods to analyze the properties of Poisson-Voronoi cells in three dimensions, revealing deviations from linear neighbor-face laws and providing asymptotic probability estimates.

Contribution

It develops asymptotic expansions for neighbor-face probabilities and face counts in 3D Poisson-Voronoi tessellations, extending 2D heuristic approaches.

Findings

Poisson-Voronoi cells violate Aboav's linear law in 3D

Asymptotic formulas for face probabilities and neighbor counts

Simulation data remains inconclusive for validation

Abstract

We consider the d-dimensional Poisson-Voronoi tessellation and investigate the applicability of heuristic methods developed recently for two dimensions. Let p_n(d) be the probability that a cell have n neighbors (be `n-faced') and m_n(d) the average facedness of a cell adjacent to an n-faced cell. We obtain the leading order terms of the asymptotic large-n expansions for p_n(d) and m_n(3). It appears that, just as in dimension two, the Poisson-Voronoi tessellation violates Aboav's `linear law' also in dimension three. A confrontation of this statement with existing Monte Carlo work remains inconclusive. However, simulations upgraded to the level of present-day computer capacity will in principle be able to confirm (or invalidate) our theory.