Transport anisotropy and percolation in the two-dimensional random-hopping model

We consider hopping transport on an anisotropic two-dimensional square lattice. The displacements parallel to one axis are governed by uniform, nearest-neighbor hopping rates c, while the displacements parallel to the other axis are governed by static but spatially fluctuating rates $w_{\mathrm{n}}$. Adapting a new class of generating functions recently introduced for the random-trapping problem, we are able to obtain expressions for the mean-square displacement in the fluctuating direction through an exact decoupling of the effects due to displacements in the uniform direction. The resulting expressions for the low-frequency diffusion coefficient D(ɛ) are exact in the limits c→0 [D(0)=〈1/w${〉}^{\mathrm{-}1}$] and c→∞ [D(0)=〈w〉]. Moreover, when the condition of long-time isotropy is imposed we obtain expressions which are, to lowest order in the fluctuations, identical to results obtained in the effective-medium approximation for the square lattice with fluctuating rates in both directions. The present method offers the possibility of systematic improvements to the effective-medium results for the dc conductivity and frequency corrections.