Diffusion-limited aggregation as branched growth

TL;DR

This paper develops a first-principles theoretical framework for diffusion-limited aggregation in two dimensions, predicting cluster properties and fractal dimensions consistent with numerical results.

Contribution

It introduces a renormalized mean-field approximation to analyze branch competition and derive cluster dimensionality and multifractal exponents.

Findings

Cluster dimensionality D ≈ 1.66

Multifractal exponent τ(3) = D

Agreement with numerical and scaling law results

Abstract

I present a first-principles theory of diffusion-limited aggregation in two dimensions. A renormalized mean-field approximation gives the form of the unstable manifold for branch competition, following the method of Halsey and Leibig [Phys. Rev. A {\bf 46}, 7793 (1992)]. This leads to a result for the cluster dimensionality, D \approx 1.66, which is close to numerically obtained values. In addition, the multifractal exponent \tau(3) = D in this theory, in agreement with a proposed `electrostatic' scaling law.