Scaling exponent of the maximum growth probability in diffusion-limited aggregation

TL;DR

This paper accurately measures the minimum growth exponent in diffusion-limited aggregation and tests a longstanding conjecture, finding it to be incorrect based on precise numerical results.

Contribution

The study provides the first precise measurement of min in DLA, conclusively testing and disproving the Turkevich-Scher conjecture.

Findings

The fractal dimension D is 1.713.

The minimum growth exponent min is 0.665.

The Turkevich-Scher conjecture does not hold for DLA.

Abstract

An early (and influential) scaling relation in the multifractal theory of Diffusion Limited Aggregation(DLA) is the Turkevich-Scher conjecture that relates the exponent \alpha_{min} that characterizes the ``hottest'' region of the harmonic measure and the fractal dimension D of the cluster, i.e. D=1+\alpha_{min}. Due to lack of accurate direct measurements of both D and \alpha_{min} this conjecture could never be put to serious test. Using the method of iterated conformal maps D was recently determined as D=1.713+-0.003. In this Letter we determine \alpha_{min} accurately, with the result \alpha_{min}=0.665+-0.004. We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.