Bias-free simulation of diffusion-limited aggregation on a square lattice

TL;DR

This paper presents a bias-free simulation algorithm for diffusion-limited aggregation on a square lattice, enabling large-scale cluster growth and analysis of anisotropic shapes and fractal dimensions.

Contribution

The authors introduce a novel algorithm that minimizes systematic errors in lattice DLA simulations, allowing for accurate large-scale cluster growth and analysis.

Findings

Lattice DLA clusters grow into anisotropic shapes influenced by lattice anisotropy.

Fractal dimension evolves from 1.71 to 1.5 as clusters become more anisotropic.

The new algorithm reduces simulation bias to below 10^{-12}.

Abstract

We identify sources of systematic error in traditional simulations of the Witten-Sander model of diffusion-limited aggregation (DLA) on a square lattice. We present an algorithm that reduces these biases to below $10^{-12}$. We grow clusters of $10^8$ particles on $65536\times 65536$ lattices. We verify that lattice DLA clusters inevitably grow into anisotropic shapes, dictated by the anisotropy of the aggregation process. We verify that the fractal dimension evolves from the continuum DLA value, $D=1.71$, for small disk-shaped clusters, towards Kesten's bound of $D=3/2$ for highly anisotropic clusters with long protruding arms.