Scaling and Crossovers in Diffusion Limited Aggregation

TL;DR

This paper investigates the scaling behavior of DLA clusters, proposing that observed anomalies are due to a slow crossover, supported by analytical and numerical evidence, and identifies a universal crossover exponent.

Contribution

It introduces a new conjecture that anomalous scaling in DLA is caused by a slow crossover, supported by analytical and numerical analysis, and suggests a unified crossover exponent.

Findings

Identifies a universal crossover exponent of -0.3 for DLA lengths.

Provides analytical support for the scaling of penetration depth.

Numerical evidence from Laurent coefficients confirms the crossover behavior.

Abstract

We discuss the scaling of characteristic lengths in diffusion limited aggregation (DLA) clusters in light of recent developments using conformal maps. We are led to the conjecture that the apparently anomalous scaling of lengths is due to one slow crossover. This is supported by an analytical argument for the scaling of the penetration depth of newly arrived random walkers, and by numerical evidence on the Laurent coefficients which uniquely determine each cluster. We find a single crossover exponent of -0.3 for all the characteristic lengths in DLA. This gives a hint about the structure of the renormalization group for this problem.