Efficient Green’s-function approach to finding the currents in a random resistor network

Using Green’s functions, we reformulate Kirchhoff’s laws for a two-component random resistor network in which a fraction p of the resistors has conductance ${\mathrm{\sigma }}_{\mathrm{-}}$ and the remainder have conductance ${\mathrm{\sigma }}_{+}$. In this Green’s-function formulation (GFF), the current correlation between any two resistors in the network is explicitly taken into account. The GFF yields a linear system equivalent to Kirchhoff’s laws but with a smaller number of variables. In the dilute case (p≪1), the voltages can be calculated directly with very high speed using the GFF. For general p, a variety of algorithms can be used to solve the GFF linear system. We present the technical details of solving the GFF linear system using the conjugate gradient method (method A). Our extensive numerical work shows that method A consistently requires fewer iterations than solving Kirchhoff’s laws directly using the conjugate gradient method (method B). For example, for a 128×128 grid with p≥0.65 and ${\mathrm{\sigma }}_{\mathrm{-}}$/${\mathrm{\sigma }}_{+}$≤${10}^{\mathrm{-}4}$, the number of iterations needed to achieve a precision of ${10}^{\mathrm{-}10}$ is more than 100 times smaller in method A than in method B.