Localization of Growth Sites in DLA Clusters: Multifractality and Multiscaling

TL;DR

This paper investigates the multifractal and multiscaling properties of DLA clusters by analyzing the distribution of growth probabilities and site locations, revealing fixed patterns for active sites and uniformity for inactive ones.

Contribution

It introduces a joint distribution framework to connect multifractality and multiscaling in DLA, highlighting fixed site locations for active growth sites.

Findings

Active growth sites are localized around a fixed radius.

Inactive sites with zero growth probability are uniformly distributed.

The joint distribution links multifractal and multiscaling behaviors.

Abstract

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.