Analytic solution of the growth-site probability distribution for structural models of diffusion-limited aggregation

We present an analytic solution of the growth-site probability distribution for a family of hierarchical models for the structure of diffusion-limited aggregation (DLA) clusters. These models are characterized by self-similar voids that are delineated by narrow channels. The growth-site probability distributions for all the models are shown to have the same form, n(α,M)∼exp{-(A/lnM)[α-${\mathrm{\alpha }}_0$(M)${]}^2$}, where n(α,M)dα is the number of growth sites with α${\mathit{p}}_{\mathit{i}}$/lnMdα, ${\mathit{p}}_{\mathit{i}}$ is the growth probability at site i, M is the cluster mass, ${\mathrm{\alpha }}_0$(M)==B lnM, and A,B are constants. We find the same form of the distribution for all members of the family of models, suggesting the possibility that it is a consequence of the channels and self-similar voids, and is independent of other details of the model. Our result is in accord with the recent calculations for DLA clusters by Schwarzer et al. [Phys. Rev. A 43, 1134 (1991)].