Multifractal Structure of the Harmonic Measure of Diffusion Limited Aggregates

Abstract

The method of iterated conformal maps allows to study the harmonic measure of Diffusion Limited Aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as $10^{-35}$, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*$ is of the order of -0.2. In the language of $f(\alpha)$ this means that $\alpha_{max}$ is finite. The $f(\alpha)$ curve loses analyticity (the phenomenon of "phase transition") at $\alpha_{max}$ and a finite value of $f(\alpha_{max})$. We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of $q^*$ and $f(\alpha_{max})$. We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.