Zipf's law for fractal voids and a new void-finder

Abstract

Voids are a prominent feature of fractal point distributions but there is no precise definition of what is a void (except in one dimension). Here we propose a definition of voids that uses methods of discrete stochastic geometry, in particular, Delaunay and Voronoi tessellations, and we construct a new algorithm to search for voids in a point set. We find and rank-order the voids of suitable examples of fractal point sets in one and two dimensions to test whether Zipf's power-law holds. We conclude affirmatively and, furthermore, that the rank-ordering of voids conveys similar information to the number-radius function, as regards the scaling regime and the transition to homogeneity. So it is an alternative tool in the analysis of fractal point distributions with crossover to homogeneity and, in particular, of the distribution of galaxies.