Diffusion Limited Aggregation with Power-Law Pinning

TL;DR

This paper investigates how a power-law decaying threshold affects Laplacian growth patterns, revealing a pinning transition at gamma=1/2 and characterizing the resulting fractal structures using multifractal analysis.

Contribution

It introduces a stochastic conformal mapping approach to analyze growth patterns with power-law pinning, identifying a critical transition point and deriving analytic expressions for fractal dimensions.

Findings

Growth patterns match DLA for gamma > 1

Lower fractal dimension for gamma < 1

Pinning transition occurs at gamma = 1/2

Abstract

Using stochastic conformal mapping techniques we study the patterns emerging from Laplacian growth with a power-law decaying threshold for growth $R_N^{-\gamma}$ (where $R_N$ is the radius of the $N-$ particle cluster). For $\gamma > 1$ the growth pattern is in the same universality class as diffusion limited aggregation (DLA) growth, while for $\gamma < 1$ the resulting patterns have a lower fractal dimension $D(\gamma)$ than a DLA cluster due to the enhancement of growth at the hot tips of the developing pattern. Our results indicate that a pinning transition occurs at $\gamma = 1/2$, significantly smaller than might be expected from the lower bound $\alpha_{min} \simeq 0.67$ of multifractal spectrum of DLA. This limiting case shows that the most singular tips in the pruned cluster now correspond to those expected for a purely one-dimensional line. Using multifractal analysis, analytic expressions are established for $D(\gamma)$ both close to the breakdown of DLA universality class, i.e., $\gamma \lesssim 1$, and close to the pinning transition, i.e., $\gamma \gtrsim 1/2$.