Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice

Abstract

The dynamics of random sequential partitioning of a square into ever smaller mutually exclusive rectangular blocks, which we call weighted planar stochastic lattice (WPSL), is governed by infinitely many conservation laws. Recently, we have shown one of the infinitely many conservation laws can be used as multifractal measure. In this article, we show that except the conservation of total area, each of the infinitely many non-trivial conservation laws is actually a multifractal measure and hence WPSL is a multi-multifractal. We then look at the block size distribution function and find that it exhibits dynamic scaling revealing that the spatial patterns of WPSL of different sizes are statistically self-similar. Besides, we investigate how the mean area ⟨A⟩k of blocks with k neighbours and the average number of neighbours mk of a typical cell that neighbours a k-sided block behaves with k. These results suggest that the Lewis law, ⟨A⟩k∝k, is obeyed for up to k ≈ 8 and the Aboav–Weaire law, kmk∝k, is violated for the entire range of k.