Scaling behavior of diffusion on percolation clusters

Abstract

A scaling analysis is performed on Monte Carlo simulations of random walks on percolation clusters both above and below the threshold $p_c$. The average diffusion constant is described by a single scaling function in which the crossover from an algebraic decay (in time) near $p_c$ to the asymptotic behavior above or below it occurs at time $t_{\mathrm{cross}}\propto {|p-p_c|}^{-(2\mathrm{}\nu -\beta +\mu )}$. The value of the percolation conductivity exponent $\mu$ is found to be 1.05 ±0.05 for two-dimensional systems and 1.5 ±0.1 for three dimensions.