Distribution-induced non-universality of the percolation conductivity exponents

The power law which characterises the behaviour of the conductivity of an inhomogeneous conductor near the percolation threshold should generally be independent of the distribution from which the conducting elements are chosen. Some counterexamples, in which a sufficiently anomalous distribution can alter the conduction threshold exponents, are exhibited. Specifically, it is claimed that a network randomly composed of insulating bonds d( sigma =0) and bonds chosen from a distribution behaving as sigma ^-alpha for small sigma will give the usual exponent t for alpha <0, but that in the case 0<or= alpha <1 the excess of small conductances alters the exponent to T=t+ alpha /(1- alpha ). Similarly, a network whose bonds are randomly superconducting ( sigma = infinity ) or chosen from a distribution whose large- sigma behaviour is sigma ^-beta will give the usual s exponent for beta >2, but in the case 1< beta <or=2 the excess of large conductances alters the exponent to S=s+(2- beta )/( beta -1).