Linear Relationship Statistics in Diffusion Limited Aggregation

TL;DR

This paper demonstrates that key surface parameters in two-dimensional diffusion limited aggregation grow linearly with cluster size, revealing proportional relationships and fractal properties of the cluster structure.

Contribution

It introduces a microscopic attachment procedure and establishes linear relationships among various surface parameters and the cluster size in DLA.

Findings

Surface parameters grow linearly with cluster size

Fractal dimension of red sites equals that of the cluster

Linear relationships between dead sites, red sites, and cluster size

Abstract

We show that various surface parameters in two-dimensional diffusion limited aggregation (DLA) grow linearly with the number of particles. We find the ratio of the average length of the perimeter and the accessible perimeter of a DLA cluster together with its external perimeters to the cluster size, and define a microscopic schematic procedure for attachment of an incident new particle to the cluster. We measure the fractal dimension of the red sites (i.e., the sites upon cutting each of them splits the cluster) equal to that of the DLA cluster. It is also shown that the average number of the dead sites and the average number of the red sites have linear relationships with the cluster size.