Self-similarity of the loop structure of diffusion-limited aggregates

Abstract

The structure of fjords in diffusion-limited aggregation (DLA) clusters can be desribed in terms of the loop size distribution nR(x) which is the normalised number of loops with a neck to depth ratio x within a circle of radius R centered at the origin of the cluster. We find from the numerical study of very large off-lattice aggregates that nR(x) converges quickly to a limiting distribution with a well-defined smallest ratio xmin larger than zero indicating the self-similarity of the loop structures, one does not expect a phase transition in multifractal spectrum of growth probabilities of typical DLA clusters generated on the plain. Our study is essentially statistical and we cannot rule out the possibility of such 'rare events' (e.g. the occurence of a few loops with anomalously small x) which may result in a qualitatively different behavior concerning the multifractal spectrum.