Conductivity of random resistor networks: An investigation of the accuracy of the effective-medium approximation.

Abstract

We investigate the electrical conductivity of disordered random resistor networks. A systematic perturbative weak-disorder expansion of the conductivity is derived, in terms of the moments of the probability distribution of the random-bond conductances. This diagrammatic technique applies to any distribution of (possibly complex) conductances, and any regular lattice of arbitrary dimensionality d. Explicit quantitative results are given, up to the sixth order of perturbation theory, for (hyper)cubic lattices in all dimensionalities of physical interest, and compared with the predictions of the effective-medium approximation (EMA). The conductivity generically departs from the EMA formula at the fourth order of perturbation theory. On the square lattice, due to the duality symmetry, the EMA starts to be incorrect only at the fifth order. In all the situations considered, the discrepancy between the EMA prediction and the exact conductivity is affected by a very small numerical factor. The limit of a large dimensionality d is also investigated. The conductivity is shown to have a systematic 1/d expansion, three terms of which are given explicitly. The discrepancy with the EMA formula is again very weak, and starts with the 1/${\mathit{d}}^{3}$ terms. This study yields, therefore, a quantitative understanding of the currently observed fact that the EMA prediction is very accurate for a large class of conductance distributions.