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 gᵢ=g₀exp(λxᵢ), where xᵢ is a random variable. In the limit λ→∞, we may write the network conductivity as σ=Ca^2-dgcλ^-y where a is the lattice constant, y a critical exponent, C a constant, and gc the percolation conductance. We derive rigorous bounds to σ and we present evidence that supports the hypothesis that $y=0$ for all two-dimensional lattices. Numerical results for a $d=3$ simple-cubic lattice are presented.