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 wn. 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(ε) are exact in the limits c →0 [D(0)=1/w-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.
K. P. Kundu et al., "Transport Anisotropy And Percolation In The Two-Dimensional Random-Hopping Model," Physical Review B (Condensed Matter), vol. 35, no. 7, pp. 3468-3477, American Physical Society (APS), Mar 1987.
The definitive version is available at https://doi.org/10.1103/PhysRevB.35.3468
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