Conductance distributions in random resistor networks. Self-averaging and disorder lengths
Abstract
The self-averaging properties of the conductance g are explored in random resistor networks (RRN) with a broad distribution of bond strengths P(g)∼g μ-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 μ. For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit μ→0. A disorder length ξD is identified, beyond which the system is effectively homogeneous. This length scale diverges as ξD∼∣µ∣-v (ν is the regular percolation correlation length exponent) when the microscopic distribution of conductors is exponentially wide (μ→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↔μ. 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 µc as ∣µ-∣-z with z∼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.
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- April 1994
- DOI:
- 10.1007/BF02186283
- arXiv:
- arXiv:cond-mat/9405010
- Bibcode:
- 1994JSP....75..135A
- Keywords:
-
- Resistor networks;
- hierarchical lattices;
- disorder;
- probability distributions;
- Condensed Matter
- E-Print:
- 16 pages plain TeX file, 6 figures available upon request.IBC-1603-012