Branched Growth with $\eta \approx 4$ Walkers

Abstract

Diffusion-limited aggregation has a natural generalization to the "$\eta$-models", in which $\eta$ random walkers must arrive at a point on the cluster surface in order for growth to occur. It has recently been proposed that in spatial dimensionality $d=2$, there is an upper critical ${\eta }_c=4$ above which the fractal dimensionality of the clusters is D=1. I compute the first order correction to $D$ for $\eta <4$, obtaining $D=1+1/2(4-\eta )$. The methods used can also determine multifractal dimensions to first order in $4-\eta$.

6 pages, 1 figure

References in corpus (3)

• Laplacian growth as one-dimensional turbulence
• Diffusion-limited aggregation as branched growth
• Multifractal Dimensions for Branched Growth

Cited by in corpus (3)

• Dimensions, Maximal Growth Sites and Optimization in the Dielectric Breakdown Model
• Growth by Random Walker Sampling, and Scaling of the Dielectric Breakdown Model
• Theory of Diffusion Controlled Growth