Some rigorous results for random planar conductance networks

Abstract

We consider planar (two-terminal) networks $Z$ of conductances ${\sigma }_{\mathrm{ij}}$ which are independently distributed according to a probability density $\rho \left(\sigma \right)$. The conductance $\overline{\sigma }$ of $Z$ is then distributed according to a probability density $R\left(\overline{\sigma }\right)$ which is determined by $\rho$ and by the graph of $Z$. We derive exact relations between the probability densities $R$ and $R^{*}$ for the conductances of two dual random networks $Z$ and $Z^{*}$, and discuss several applications. If, for infinite regular lattices, it is assumed that $\overline{\sigma }$ is dispersion free, we can prove that the effective-medium result for $\overline{\sigma }$ satisfies all our exact relations.