Diffusion on percolation clusters at criticality

The concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters. The authors find from Monte Carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three-dimensional space is D=3.3+or-0.1 where the size of the cluster is restricted to be larger than the span of the walk, and is D’=3.9+or-0.1 for a walk on clusters not subject to this restriction. For two-dimensional space they find D approximately=D’ approximately=2.7+or-P0.1. The exponent D (and D’) is related to the scaling of the average length R of N steps via R^Dvaries as N. The fracton dimensionality which is related to the density of states was found to be 1.26+or-0.1. These results are in good agreement with the predictions of Alexander and Orbach (1982).