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 $\{ p_i\}$, where $p_i$ is the probability that the perimeter site $i$ will be the next to grow. We introduce the joint distribution $N(\alpha, x, M)$, where $N(\alpha,x,M)d\alpha dx$ is the number of perimeter sites with $\alpha$-values in the range $\alpha\le \alpha_i \le \alpha+d\alpha$ (``$\alpha$-sites'') and located in the annulus [x, x+dx] around the cluster seed. Here, $\alpha_i \equiv -\ln p_i / \ln R$ if $p_i>0$, $x\equiv r_i/R$, and $r_i$ is the distance of site $i$ from the seed of the DLA cluster. We use $N(\alpha,x,M)$ to relate multifractal and multiscaling properties of DLA. In particular, we find that for large $M$ the location of the $\alpha$-sites is peaked around a fixed value $\bar x(\alpha)$; in contrast, the perimeter sites with $p_i=0$ are uniformly distributed over the DLA cluster.