How anisotropy beats fractality in two-dimensional on-lattice DLA growth

TL;DR

This study reveals that two-dimensional lattice DLA clusters grow anisotropically with a dominant axis, leading to a deterministic one-dimensional scaling limit, while maintaining a non-trivial fractal structure due to fluctuations around this backbone.

Contribution

It demonstrates that anisotropy dominates DLA growth on a lattice, challenging the traditional fractal perspective and highlighting the importance of fluctuations in the cluster structure.

Findings

DLA clusters grow fastest along the axes.

The scaling limit is deterministic and one-dimensional.

The cluster retains a non-trivial fractal dimension.

Abstract

We study the fractal structure of Diffusion-Limited Aggregation (DLA) clusters on the square lattice by extensive numerical simulations (with clusters having up to $10^8$ particles). We observe that DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along the axes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster it has non-trivial box-counting fractal dimension which corresponds to the overall growth rate which, in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of the lattice DLA should be understood in terms of fluctuations around one-dimensional backbone of the cluster.