Percolationlike exponent for the conductivity of highly disordered resistor networks

It has been proposed that the conductivity of highly disordered resistor networks with bond conductances $g_i=g_0\exp \left(\lambda x_i\right)$ and $\lambda \to \infty$ for fixed $P\left(x\right)\mathrm{}$ behaves as $\Sigma \sim g_c{\lambda }^{-y}$ (where $g_c$ is the percolation conductance). We argue that $y=(d-2\mathrm{})\mathrm{}\nu \mathrm{}(d<\mathrm{}6\mathrm{})\mathrm{}$ and $y=2\mathrm{}(d\ge 6)$, where $\nu$ is the percolation correlation-length exponent. This allows us to recover the "nonuniversal" conductivity exponents of percolation with broad distributions. A scaling form for superconductor-normal-insulator mixtures is proposed. Similar arguments apply to magnetic systems, diffusion, and directed percolation.