Anomalous Diffusion on Percolating Clusters

The mean square distance reached after $t$ random-walk steps on a percolating cluster is shown to behave as $t^{\frac{2}{(2+\theta )}}$ (instead of $t$) for short times (for which the cluster is seen to be self-similar). The exponent $\theta$ is related to that describing the dc conductivity near percolation. Averaging over all clusters yields the ac conductivity and dielectric constant near percolation in the high-frequency limit, when the polarization of the medium is unimportant.