Number of branches in diffusion-limited aggregates: The skeleton

TL;DR

This paper introduces the skeleton algorithm to analyze the main branching structure of large diffusion-limited aggregation clusters across different dimensions, revealing dimension-dependent ramification properties and scaling behaviors.

Contribution

The study applies the skeleton algorithm to large off-lattice DLA clusters in multiple dimensions, uncovering how the number of main branches varies with distance and dimension, and identifying scaling corrections.

Findings

In 2D, the number of main branches levels off at about 7.5.

In 3D and 4D, the number of branches continues to grow outward.

In 2D, strong logarithmic corrections to scaling are observed.

Abstract

We develop the skeleton algorithm to define the number of main branches $N_b$ of diffusion-limited aggregation (DLA) clusters. The skeleton algorithm provides a systematic way to remove dangling side branches of the DLA cluster and has successfully been applied to study the ramification properties of percolation. We study the skeleton of comparatively large ($\approx 10^6$ sites) off-lattice DLA clusters in two, three and four spatial dimensions. We find that initially with increasing distance from the cluster seed the number of branches increases in all dimensions. In two dimensions, the increase in the number of branches levels off at larger distances, indicating a fixed number of $N_b = 7.5\pm 1.5$ main branches of DLA. In contrast, in three and four dimensions, the skeleton continues to ramify strongly as one proceeds from the cluster center outward, and we find no indication of a constant number of main branches. Likewise, we find no indication for a fixed $N_b$ in a study of DLA on the Cayley tree. In two dimensions, we find strong corrections to scaling of logarithmic character, which can help to explain recently reported deviations from self-similar behavior.