Combinatorial and approximative analyses in a spatially random division process

TL;DR

This paper introduces a new analytical approach to understand the distributions of fragment sizes in a spatial division process by modeling it as a fractal-like hierarchical network and deriving combinatorial and continuous approximations.

Contribution

It presents a novel analytical framework for describing size distributions in spatial division processes using a Markov chain model and combinatorial analysis.

Findings

Derived a combinatorial analytical form for area distribution.

Showed good numerical fit of the model to actual data.

Established a connection between random divisions and fractal-like structures.

Abstract

For a spatial characteristic, there exist commonly fat-tail frequency distributions of fragment-size and -mass of glass, areas enclosed by city roads, and pore size/volume in random packings. In order to give a new analytical approach for the distributions, we consider a simple model which constructs a fractal-like hierarchical network based on random divisions of rectangles. The stochastic process makes a Markov chain and corresponds to directional random walks with splitting into four particles. We derive a combinatorial analytical form and its continuous approximation for the distribution of rectangle areas, and numerically show a good fitting with the actual distribution in the averaging behavior of the divisions.