Planar Voronoi cells : the violation of Aboav's law explained

TL;DR

This paper derives an exact large-N expansion for the average sidedness of cells neighboring an n-sided cell in planar Voronoi tessellations, explaining the observed curvature in Aboav's law and predicting similar behavior in experimental data.

Contribution

It provides the first theoretical explanation for the curvature in Aboav's law using an exact large-N expansion of m_n in Voronoi cells.

Findings

Derives an exact large-N expansion: m_n=4+3(π/n)^{1/2}+...

Explains the downward curvature in nm_n data of Voronoi tessellations.

Predicts similar curvature should be observable in high-resolution experimental data.

Abstract

In planar cellular systems $m\_n$ denotes the average sidedness of a cell neighboring an $n$-sided cell. Aboav's empirical law states that $nm\_n$ is linear in $n$. A downward curvature is nevertheless observed in the numerical $nm\_n$ data of the Random Voronoi Froth. The exact large-N expansion of $m\_n$ obtained in the present work, {\it viz.} $m\_n=4+3(\pi/n)^{{1/2}}+...$, explains this curvature. Its inverse square root dependence on $n$ sets a new theoretical paradigm. Similar curved behavior may be expected, and must indeed be looked for, in experimental data of sufficiently high resolution. We argue that it occurs, in particular, in diffusion-limited colloidal aggregation on the basis of recent simulation data due to Fern\'andez-Toledano {\it et al.} [{\it Phys. Rev. E} {\bf 71}, 041401 (2005)] and earlier experimental results by Earnshaw and Robinson [{\it Phys. Rev. Lett.} {\bf 72}, 3682 (1994)].