Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results

TL;DR

This paper provides a detailed asymptotic analysis of the large-sided planar Poisson-Voronoi cell, deriving probability laws, correlations, and shape characteristics as the number of sides grows large.

Contribution

It introduces exact asymptotic expansions for the probability and shape statistics of large n-sided Poisson-Voronoi cells, including correlations and perimeter deviations.

Findings

The n-sided cell approaches a circular shape with radius proportional to √n.

The probability of n sides has an explicit asymptotic expansion.

Perimeter deviations follow a Gaussian noise model.

Abstract

We achieve a detailed understanding of the $n$-sided planar Poisson-Voronoi cell in the limit of large $n$. Let ${p}\_n$ be the probability for a cell to have $n$ sides. We construct the asymptotic expansion of $\log {p}\_n$ up to terms that vanish as $n\to\infty$. We obtain the statistics of the lengths of the perimeter segments and of the angles between adjoining segments: to leading order as $n\to\infty$, and after appropriate scaling, these become independent random variables whose laws we determine; and to next order in $1/n$ they have nontrivial long range correlations whose expressions we provide. The $n$-sided cell tends towards a circle of radius $(n/4\pi\lambda)^{\half}$, where $\lambda$ is the cell density; hence Lewis' law for the average area $A\_n$ of the $n$-sided cell behaves as $A\_n \simeq cn/\lambda$ with $c=1/4$. For $n\to\infty$ the cell perimeter, expressed as a function $R(\phi)$ of the polar angle $\phi$, satisfies $d^2 R/d\phi^2 = F(\phi)$, where $F$ is known Gaussian noise; we deduce from it the probability law for the perimeter's long wavelength deviations from circularity. Many other quantities related to the asymptotic cell shape become accessible to calculation.