Localization of Growth Sites in DLA Clusters: Multifractality and Multiscaling

The growth of a diffusion limited aggregation (DLA) cluster with mass M and radius of gyration R is described by a set of growth probabilities {pi}, where pi is the probability that the perimeter site i will be the next to grow. We introduce the joint distribution N(α, x,M), where N(α, x,M)dαdx is the number of perimeter sites with α-values in the range α ≤ αi ≤ α + dα (“αsites”) and located in the annulus [x, x+dx] around the cluster seed. Here, αi ≡ − ln pi/ lnR if pi > 0, x ≡ ri/R, and ri is the distance of site i from the seed of the DLA cluster. We use N(α, x,M) to relate multifractal and multiscaling properties of DLA. In particular, we find that for large M the location of the α-sites is peaked around a fixed value x̄(α); in contrast, the perimeter sites with pi = 0 are uniformly distributed over the DLA cluster. 61.43.Hv, 68.70+w, 05.40+j, 81.10.-h Typeset using REVTEX 1