Random-walk approach to the two-component random-conductor mixture: Perturbing away from the perfect random resistor network and random superconducting-network limits

Abstract

We develop a random-walk approach that permits one to study novel physical phenomena that occur in the random two-component mixture of good and poor conductors. Our work significantly generalizes the previous body of knowledge on the random resistor network (pure ``ant'' limit) in which the poor-conductor species has infinite resistance, and the random superconductor network (pure ``termite'' limit) in which the good-conductor species has zero resistance. We find that for any fixed value of the concentration p of good conductors, we can map a system that is nearer the ant limit to one that is nearer the termite limit. Specifically, we find ${R}^{2}$(1,h,t/h)=${R}^{2}$(${h}^{\mathrm{\ensuremath{-}}1}$, 1,t) where ${R}^{2}$(${\ensuremath{\sigma}}_{a}$,${\ensuremath{\sigma}}_{b}$,t) is the mean-square displacement of the walker, and h=${\ensuremath{\sigma}}_{b}$/${\ensuremath{\sigma}}_{a}$ is the ratio of the conductivity of the poor and good components. This exact transformation permits one to develop a scaling theory for the general two-component case, which reduces to the known results for the ant limit and predicts dramatically new behavior for the termite limit. We test the scaling predictions extensively by Monte Carlo simulation methods. Finally, we develop an analogy with a simple magnetic system, in which the role of the magnetic field is played by the conductivity ratio h.