CONDUCTANCE DISTRIBUTIONS IN RANDOM RESISTOR NETWORKS - SELF-AVERAGING AND DISORDER LENGTHS

The self-averaging properties of the conductance g are explored in random resistor networks (RRN) with a broad distribution of bond strengths P(g) approximately g(mu-1). The RRN problem is cast in terms of simple combinations of random variables on hierarchical lattices. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of size L and the distribution tail strength parameter mu. For networks above the percolation threshold, convergence to a Gaussian basin is always the case. except in the limit mu --> 0. A disorder length xi(D) is identified, beyond which the system is effectively homogeneous. This length scale diverges as xi(p) approximately \mu\-nu (nu is the regular percolation correlation length exponent) when the microscopic distribution of conductors is exponentially wide (mu --> 0). This implies that exactly the same critical behavior can be induced by geometrical disorder and by Strong bond disorder with the bond occupation probability p <-- --> mu. We find that only lattices at the percolation threshold have renormalized probability distributions in a Levy-like basin. At the percolation threshold the disorder length diverges at a critical tail strength mu(c) as \mu - mu(c)\-z with z approximately 3.2 +/- 0.1, a new exponent. Critical path analysis is used in a generalized form to give the macroscopic conductance in the case of lattices above p(c).