Some rigorous results for random planar conductance networks
Jakob Bernasconi, William R. Schneider, Harold J. Wiesmann · Physical review. B, Solid state · 1977
We consider planar (two-terminal) networks $Z$ of conductances ${\ensuremath{\sigma}}_{\mathrm{ij}}$ which are independently distributed according to a probability density $\ensuremath{\rho}(\ensuremath{\sigma})$. The conductance $\overline{\ensuremath{\sigma}}$ of $Z$ is then distributed according to a probability density $R(\overline{\ensuremath{\sigma}})$ which is determined by $\ensuremath{\rho}$ 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 $\overline{\ensuremath{\sigma}}$ is dispersion free, we can prove that the effective-medium result for $\overline{\ensuremath{\sigma}}$ satisfies all our exact relations.