Limit distributions in random resistor networks

The question of attraction to stable limit distributions in random resistor networks (RRNs) is explored numerically. Transport in networks with power law distributions of conductances of the form P(g) = |μ|g^μ-1 are considered. Distributions of equivalent conductances are estimated on hierarchical lattices as a function of size L and the parameter μ. We find that only lattices at the percolation threshold can support transport in a Levy-like basin. For networks above the percolation threshold, convergence to a Gaussian basin is always the case, and a disorder length ξ_D is identified, beyond which the system is effectively homogeneous. This length scale diverges, when the microscopic distribution of conductors is exponentially wide (μ→0), as ξ_D∼|μ|^-1.6-0.1.