Asymptotic statistics of the n-sided planar Voronoi cell: II. Heuristics

TL;DR

This paper presents heuristic explanations for properties of large-sided cells in planar Poisson-Voronoi tessellations, including phase space analysis and an application to Gabriel neighbors, offering insights into their asymptotic behavior.

Contribution

It introduces heuristic methods to explain and approximate the properties of large n-sided Voronoi cells, complementing previous rigorous results.

Findings

Heuristic entropy-based arguments for large n-sided cells.

Simplified phase space integral evaluation for probability p_n.

Calculation of expected Gabriel neighbors for large n.

Abstract

We develop a set of heuristic arguments to explain several results on planar Poisson-Voronoi tessellations that were derived earlier at the cost of considerable mathematical effort. The results concern Voronoi cells having a large number n of sides. The arguments start from an entropy balance applied to the arrangement of n neighbors around a central cell. It is followed by a simplified evaluation of the phase space integral for the probability p_n that an arbitrary cell be n-sided. The limitations of the arguments are indicated. As a new application we calculate the expected number of Gabriel (or full) neighbors of an n-sided cell in the large-n limit.