Anisotropic Diffusion Limited Aggregation

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.

arXiv comment6 pages, 4 figures, submitted to Phys. Rev. E

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• Laplacian Growth and Diffusion Limited Aggregation: different universality classes
• Renormalization Theory of Stochastic Growth
• Laplacian growth with separately controlled noise and anisotropy
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Cited by in corpus (2)

• Scaling limits of anisotropic Hastings-Levitov clusters
• Fractal Structure of Hastings-Levitov Patterns Restricted in a Sector Geometry