Anisotropic generalization of Stinchcombe's solution for conductivity of random resistor network on a Bethe lattice

TL;DR

This paper extends Stinchcombe's solution to anisotropic Bethe lattices, providing an analytical approximation for the conductivity of random resistor networks that is accurate above the percolation threshold.

Contribution

It introduces an anisotropic generalization of Stinchcombe's solution using a power series expansion in inverse coordination number, improving conductivity estimates for anisotropic networks.

Findings

Analytical conductivity approximation matches numerical results above percolation threshold.

Expansion provides accurate results for resistor concentrations near the percolation point.

Comparison shows good agreement with regular lattice results by Bernasconi.

Abstract

Our study is based on the work of Stinchcombe [1974 \emph{J. Phys. C} \textbf{7} 179] and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic Bethe lattice. The structure of the Bethe lattice is assumed to represent the normal directions of the regular lattice. We calculate the anisotropic conductivity as an expansion in powers of inverse coordination number of the Bethe lattice. The expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold. We make a comparison of our analytical results with those of Bernasconi [1974 \emph{Phys. Rev. B} \textbf{9} 4575] for the regular lattice.