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

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 $(n/4πλ)^{\half}$, 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 $R(\varphi )$ of the polar angle $\varphi$, satisfies $d^{2}R/d{\varphi }^2=F(\varphi )$, 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.

54 pages, 3 figures