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

In planar cellular systems mn denotes the average sidedness of a cell neighbouring 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 nmn data of the random Voronoi froth. The exact large-n expansion of mn obtained in the present work, namely, 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 behaviour 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 Fernandez-Toledano et al (2005 Phys. Rev. E 71 041401) and earlier experimental results by Earnshaw and Robinson (1994 Phys. Rev. Lett. 72 3682).