Scaling and Multiscaling Behavior of the Perimeter of Diffusion-Limited Aggregation (DLA) Generated by the Hastings-Levitov Method

TL;DR

This paper investigates the scaling and multiscaling properties of DLA clusters generated by the Hastings-Levitov method, confirming fractal dimensions and growth exponents, and revealing complex boundary scaling behaviors.

Contribution

It provides a detailed analysis of the fractal and multiscaling behavior of DLA clusters using the Hastings-Levitov method, including new insights into boundary length scales and their fluctuations.

Findings

Fractal dimension of clusters matches analytical predictions.

Growth exponent $eta=0.557(2)$ determined from interface evolution.

Perimeter exhibits asymptotic multiscaling behavior.

Abstract

In this paper, we analyze the scaling behavior of \emph{Diffusion Limited Aggregation} (DLA) simulated by Hastings-Levitov method. We obtain the fractal dimension of the clusters by direct analysis of the geometrical patterns in a good agreement with one obtained from analytical approach. We compute the two-point density correlation function and we show that in the large-size limit, it agrees with the obtained fractal dimension. These support the statistical agreement between the patterns and DLA clusters. We also investigate the scaling properties of various length scales and their fluctuations, related to the boundary of cluster. We find that all of the length scales do not have a simple scaling with same correction to scaling exponent. The fractal dimension of the perimeter is obtained equal to that of the cluster. The growth exponent is computed from the evolution of the interface width equal to $\beta=0.557(2)$. We also show that the perimeter of DLA cluster has an asymptotic multiscaling behavior.