The Harmonic Measure of Diffusion-Limited Aggregates including Rare Events

TL;DR

This paper introduces a highly accurate sampling method to measure the harmonic measure of DLA clusters, revealing multifractal properties and phase transitions in the measure's spectrum, even for very large clusters.

Contribution

A novel biased random-walk sampling algorithm enabling precise harmonic measure measurement for larger DLA clusters than previous methods.

Findings

Measured probabilities as small as 10^(-80)

Identified a phase transition in the multifractal spectrum at alpha≈14

Determined a minimum q of D(q) at approximately 0.9

Abstract

We obtain the harmonic measure of diffusion-limited aggregate (DLA) clusters using a biased random-walk sampling technique which allows us to measure probabilities of random walkers hitting sections of clusters with unprecedented accuracy; our results include probabilities as small as 10^(-80). We find the multifractal D(q) spectrum including regions of small and negative q. Our algorithm allows us to obtain the harmonic measure for clusters more than an order of magnitude larger than those achieved using the method of iterative conformal maps, which is the previous best method. We find a phase transition in the singularity spectrum f(alpha) at alpha approximately equal to 14 and also find a minimum q of D(q), q_{min} = 0.9 plus or minus 0.05.