Random resistor network with an exponentially wide distribution of bond conductances

We use a percolation analysis to study the conductivity of a random resistor network with bond conductances $g_i=g_0\exp \left(\lambda x_i\right)$, where $x_i$ is a random variable. In the limit $\lambda \to \infty$, we may write the network conductivity as $\sigma =C{a}^{2-d}{g}_c{\lambda }^{-y}$ where $a$ is the lattice constant, $y$ a critical exponent, $C$ a constant, and $g_c$ the percolation conductance. We derive rigorous bounds to $\sigma$ and we present evidence that supports the hypothesis that $y=0\mathrm{}$ for all two-dimensional lattices. Numerical results for a $d=3\mathrm{}$ simple-cubic lattice are presented.