Critical phenomena in resistor networks

Abstract

The conduction threshold which occurs in resistor networks when a certain fraction has zero conductivity can be treated as a sort of critical point. This critical point is also relevant to mixtures of resistors with finite conductivities a and b, in the limit a<<b. A homogeneous function representation is proposed for the conductivity in the vicinity of the conduction threshold and a parametric representation is also given. This representation makes predictions for the concentration (p) and composition (a/b) dependence of such networks which are consistent with the behaviour of the Bethe lattice model recently proposed by Stinchcombe (1973, 1974). The frequency dependence of the impedance is also discussed and a correlation length which diverges at the critical point is described.