Invasion percolation and invading Eden growth on multifractal lattices

Abstract

Eden growth and invasion percolation models have been explored on square lattices with a multifractal distribution of growth probabilities. These models generate structures with a mass fractal dimensionality of 2 and surfaces which can be described in terms of self-similar fractal geometry with a fractal dimensionality larger than 1 and smaller than 2. For the Eden growth models the fractal dimensions of the total perimeter, internal hull and external hull depend on the probabilities associated with the generator of the multifractal measure. For the invasion percolation model the three fractal dimensionalities are relatively insensitive to the structure of the generator and may be universal. The distribution of growth probabilities associated with the Eden models can also be described as a fractal measure and the spectrum of singularities f( alpha ) associated with this measure has been estimated for some of these models. For the Eden growth models, the fractal dimensionalities describing the structure of total perimeter, internal hull and external hull are given by D=d( xi ) where xi is the variance of the logarithms of the probabilities used in the multiplicative generators for the multifractal growth probability measure.