Invasion percolation on substrates with correlated disorder

Abstract

Simulations have been carried out to investigate invasion percolation with trapping on two dimensional substrates with a multifractal distribution of bond threshold probabilities. Suprisingly, the spatial correlations in the bond thresholds appear not to influence or influence only slightly the fractal scaling properties. Fractal dimensionalities, D, of about 1.82 for the entire clusters, about 1.45–1.5 for the hull and about 1.4–1.45 for the outer hull were obtained for all bond invasion percolation models with and without spatial correlations. The exponent τ describing the power law distribution of trapped defender cluster sizes has a value of about 1.90 (( D + d)/ d where d is the Euclidean dimensionality). The value for D of 1.4–1.45 for the outer hull is regarded as being consistent with but not in good agreement with the theoretical value of 4 3 for ordinary percolation. The value of about 1.50 obtained for the total hull is much different from the value of 1.75 associated with ordinary percolation hulls. The difference between the value of 1.82 obtained from the entire cluster and the theoretical percolation value of vl is (1.895833) is also regarded as being significant. The simulation results also show that the fractal dimensionality for the total hull is much different for bond and site invasion percolation with trapping on substrates with uncorrelated disorder while the outer hulls have the same fractal dimensionality for both models. For the site invasion percolation model D hull ⋍ 7 4 while for the bond invasion percolation model D hull ⋍D outer hull .