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

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)].