SELF-SIMILARITY OF THE LOOP STRUCTURE OF DIFFUSION-LIMITED AGGREGATES

The structure of fjords in diffusion-limited aggregation (DLA) clusters can be described 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 centred at the origin of the cluster. The authors 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 structure. If the fjords are self-similar, i.e. they do not have long, tube-like structures, one does not expect a phase transition in the multifractal spectrum of growth probabilities of typical DLA clusters generated on the plain. Their study is essentially statistical and they cannot rule out the possibility of such 'rare events' (e.g. the occurrence of a few loops with anomalously small x) which may result in a qualitatively different behaviour concerning the multifractal spectrum.