Random walks in random environments

A renormalization-group analysis is carried out of the long-time behavior of random walks in an environment with a positionally random local drift force. It is argued that, independent of the strength of the disorder, the mean-square displacement, $〈x^2\mathrm{}\left(t\right)\mathrm{}〉$, is linear in time (i.e., diffusive) for dimensions $d\gtrsim 2$. In two dimensions, universal $\frac{t}{\ln t}$ corrections are found and for $d=2-\epsilon$, the behavior is subdiffusive with $〈x^2\mathrm{}\left(t\right)\mathrm{}〉\sim t^{1-{\epsilon }^2}$.