Motion in media with percolation thresholds

Sites on a lattice are randomly denoted "good" or "bad" with probability $p$ and ($1-p$), respectively. A particle confined to the good sites executes a random walk with nearest-neighbor hopping. We give a method for obtaining the diffusion constant $D$ for all $p$, based upon the real-space renormalization group; $D$ is seen to vanish as $p$ approaches the percolation threshold $p^{*}$. We find $D$ for all $p$ on a triangular lattice and we investigate the $p\sim p^{*}$ regime, where $D$ is found to vary as ${\xi }^{-x}f\left(\frac{\omega }{{\xi }^{-y}}\right)$, $x\sim 0.48$, $y\sim 2.48$, and $\xi$ is the correlation length. Accordingly, $D$ vanishes as ${\epsilon }^{0.65}$, $\epsilon =|p-p^{*}|$, and anomalous diffusion occurs on the percolation cluster with the mean-square displacement varying as $t^{0.81}$. Further analysis of our earlier theory of the Lorentz gas (where the scatterers define a continuum percolation problem), based upon the self-consistent repeated ring approximation, is shown also to yield the scaling form of $D$ with $D\sim {\epsilon }^{0.5}$, $\epsilon =\frac{|\rho -{\rho }^{*}|}{{\rho }^{*}}$, where $\rho$ is the reduced density, and with the mean-square displacement at ${\rho }^{*}$ increasing as $t^{\frac{2}{3}}$ in two and three dimensions.