Random walk on percolation clusters

Abstract

Random-walk simulations at and above the percolation threshold were performed for two- and three-dimensional lattices. The number of distinct sites visited on the percolating cluster at threshold obeys the "superuniversality" relation, showing a fractal spectral (fracton) dimensionality of $\frac{4}{3}$. We also observe fractal-to-Euclidean crossovers with time and with increased concentration (above threshold). The mean-squared displacement (from the origin) shows a similar behavior and its critical exponents are compared with previous work.