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

TL;DR

This paper investigates how conductance distributions in random resistor networks self-average and introduces the concept of a disorder length, revealing critical behavior and new exponents near the percolation threshold.

Contribution

It provides a numerical analysis of conductance distributions, identifies a disorder length diverging at criticality, and links geometrical and bond disorder effects.

Findings

Convergence to Gaussian conductance distribution above percolation threshold.

Disorder length diverges as a power law near criticality, with a new exponent.

Critical behavior can be induced by either geometrical or strong bond disorder.

Abstract

The self averaging properties of conductance $g$ are explored in random resistor networks with a broad distribution of bond strengths $P(g)\simg^{\mu-1}$. 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 $p_c$.