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 $\mu$. For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit $\mu$ --> 0. A {\it disorder length} ${\xi }_D$ is identified beyond which the system is effectively homogeneous. This length diverges as ${\xi }_D\sim |\mu {|}^{-\nu }$ ($\nu$ is the regular percolation correlation length exponent) as $\mu$-->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 $p$<-->$\mu$. 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 ${\mu }_c$ as $|\mu -{\mu }_c{|}^{-z}$, 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 $p_c$.

arXiv comment16 pages plain TeX file, 6 figures available upon request.IBC-1603-012

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