Eden growth on multifractal lattices

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Abstract
A modified Eden model has been investigated in which the growth probabilities are determined by a fractal measure on the underlying lattice. Both the spatial distribution and probability distribution of surface sites on the growing clusters have been investigated. For the case where the multifractal substrate is constructed using a 2*2 multiplicative generator with the probabilities P1=1, P2=R, P3=R3 and P4=R3, the spatial distribution of surface sites and the sites comprising the inner and outer hulls have a fractal geometry which can be described by dimensionalities which depend on R. For the total surface, this dimensionality converges to a value of about 1.76+or-0.01 as R to 0. For the inner and outer hulls the fractal dimensionality approaches a value of about 1.48+or-0.02.