Scaling laws and topological exponents in Voronoi tessellations of intermittent point distributions

Abstract

Voronoi tessellations of scale-invariant fractal sets are characterized by topological and metrical properties that are significantly different from those of natural cellular structures. As an example we analyze Voronoi diagrams of intermittent particle distributions generated by a directed percolation process in (2+1) dimensions. We observe that the average area of a cell increases much faster with the number of its neighbours than in natural cellular structures where Lewis' law predicts a linear behaviour. We propose and numerically verify a universal scaling law that relates shape and size of the cells in scale-invariant tessellations. A novel exponent, related to the topological properties of the tessellation, is introduced and estimated numerically.