Branched growth with eta approximately 4 walkers.
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.
Halsey
Thomas C
TC
Corporate Strategic Research, ExxonMobil Research and Engineering, Route 22 East, Annandale, New Jersey 08801, USA. thomas.c.halsey@exxonmobil.com