Diffusion and Long-Time Tails in a Two-Dimensional Site-Percolation Model

For a simple Lorentz gas, viz., a random walk on a square lattice with randomly excluded sites, we present an exact calculation of the diffusion coefficient, the static conductivity, and the long-time tail of the velocity autocorrelation function $\phi \mathrm{}\left(t\right)\mathrm{}\simeq -\frac{\beta \mathrm{}\left(c\right)\mathrm{}}{t^2}$, exact to $O\left(c^2\right)$ terms included, by making a systematic expansion in the density of impurities $c$ (fraction of excluded sites). To $O\left(c\right)\mathrm{}$ the velocity auto-correlation function is calculated exactly for all times, and shows negative correlations (cage effect).