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Conductance distributions in random resistor networks. Self-averaging and disorder lengths

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The self-averaging properties of the conductanceg are explored in random resistor networks (RRN) with a broad distribution of bond strengthsP(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 sizeL 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. Adisorder 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 probabilityp↔μ. We find that only lattices at the percolation threshold have renormalized probability distributions in aLevy-like basin. At the percolation threshold the disorder length diverges at a critical tail strength µc as ∣µ−∣−z withz∼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 abovep c.

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Angulo, R.F., Medina, E. Conductance distributions in random resistor networks. Self-averaging and disorder lengths. J Stat Phys 75, 135–151 (1994).

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Key Words

  • Resistor networks
  • hierarchical lattices
  • disorder
  • probability distributions