The perimeter of large planar Voronoi cells: a double-stranded random walk

TL;DR

This paper derives the asymptotic behavior of large planar Voronoi cells, revealing that their perimeter is governed by two biased random walks and confirming Lewis' law with a specific coefficient.

Contribution

It introduces a novel probabilistic model involving two biased random walks to describe the perimeter of large Voronoi cells and derives their asymptotic properties.

Findings

Asymptotic expansion of log probability for n-sided cells

Limit distributions for angles between vertices and perimeter segments

Proof of Lewis' law with coefficient 1/4

Abstract

Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to terms that vanish as $n\to\infty$. We show that {\it two independent biased random walks} executed by the polar angle determine the trajectory of the cell perimeter. We find the limit distribution of (i) the angle between two successive vertex vectors, and (ii) the one between two successive perimeter segments. We obtain the probability law for the perimeter's long wavelength deviations from circularity. We prove Lewis' law and show that it has coefficient 1/4.