Conductivity in percolation networks with broad distributions of resistances

Diluted resistor networks with a broad distribution of resistances are studied near the percolation threshold. A hierarchical model of the backbone of the percolation cluster is employed. Resistor networks are considered where the resistors, R, are chosen from a distribution having a power-law tail such that Prob{R>X}∼$X^{\mathrm{-}\alpha }$ as X→∞, 0<α<1. Such distributions arise naturally in con- tinuum percolation systems. The hierarchical model is studied numerically and using a renormalization-group transformation for the distribution of resistances. The conclusion is that the conductivity exponent t is the greater of $t_o$ and (d-2)ν+1/α where $t_o$ is the universal value of the conductivity exponent and ν is the correlation-length exponent. This result is in agreement with Straley’s earlier predictions [J. Phys. C 15, 2333 (1982); 15, 2343 (1982)].