The harmonic measure of diffusion-limited aggregates including rare events

¹  Department of Physics, University of Michigan - Ann Arbor, MI, 48109-1040, USA
²  Michigan Center for Theoretical Physics, University of Michigan - Ann Arbor, MI, 48109-1040, USA
³  Department of Physics and Centre for Complexity Science, University of Warwick - Coventry CV4 7AL, UK, EU
⁴  Center for the Study of Complex Systems, University of Michigan - Ann Arbor, MI, 48109-1040, USA
⁵  Department of Chemical Engineering, University of Michigan - Ann Arbor, MI, 48109-2136, USA

Corresponding author: davidada@umich.edu

Received: 2 June 2009
Accepted: 29 June 2009

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 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 at α ≈ 14 and also find a minimum q of , q_min = 0.9 ± 0.05.