Anisotropic Diffusion Limited Aggregation

TL;DR

This paper investigates how anisotropic perturbations influence diffusion limited aggregation (DLA) in two dimensions, revealing a transition from DLA to needle-like structures and analyzing the fractal dimension's dependence on anisotropy parameters.

Contribution

The study introduces a stochastic conformal mapping approach to model anisotropic effects on DLA, providing a continuous transition framework from DLA to needle-like growth patterns.

Findings

Fractal dimension decreases from ~1.71 to 1.5 with increasing anisotropy.

A parameter space (m,k) characterizes the transition from DLA to needle-like structures.

Scaling arguments match numerical results for the fractal dimension dependence.

Abstract

Using stochastic conformal mappings we study the effects of anisotropic perturbations on diffusion limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map $m$ preferred directions for growth of angular width $\sigma$ to a distribution on the unit circle which is a periodic function with $m$ peaks in $[-\pi, \pi)$ such that the width $\sigma$ of each peak scales as $\sigma \sim 1/\sqrt{k}$, where $k$ defines the ``strength'' of anisotropy along any of the $m$ chosen directions. The two parameters $(m,k)$ map out a parameter space of perturbations that allows a continuous transition from DLA (for $m=0$ or $k=0$) to $m$ needle-like fingers as $k \to \infty$. We show that at fixed $m$ the effective fractal dimension of the clusters $D(m,k)$ obtained from mass-radius scaling decreases with increasing $k$ from $D_{DLA} \simeq 1.71$ to a value bounded from below by $D_{min} = 3/2$. Scaling arguments suggest a specific form for the dependence of the fractal dimension $D(m,k)$ on $k$ for large $k$, form which compares favorably with numerical results.