Conductance Distributions in Random Resistor Networks: Self Averaging and Disorder Lengths

Abstract

The self averaging properties of conductance $g$ are explored in random resistor networks with a broad distribution of bond strengths $P(g)\simg^{μ-1}$. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of size $L$ and distribution tail parameter $μ$. For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit $μ$ --> 0. A {\it disorder length} $ξ_D$ is identified beyond which the system is effectively homogeneous. This length diverges as $ξ_D \sim |μ|^{-ν}$ ($ν$ is the regular percolation correlation length exponent) as $μ$-->0. This suggest that exactly the same critical behavior can be induced by geometrical disorder and bu strong bond disorder with the bond occupation probability $p$<-->$μ$. Only lattices at the percolation threshold have renormalized probability distribution in a {\it Levy-like} basin. At the threshold the disorder length diverges at a vritical tail strength $μ_c$ as $|μ-μ_c|^{-z}$, with $z=3.2\pm 0.1$, a new exponent. Critical path analysis is used in a generalized form to give form to give the macroscopic conductance for lattice above $p_c$.