Field-Theoretic Formulation of the Randomly Diluted Nonlinear Resistor Network

A field-theoretic formulation is used to describe the resistive properties of a randomly diluted network consisting of nonlinear conductances for which V~Ir. The nonlinear resistance R(x,x’) between sites x and x’ is expressed in terms of an analytic continuation in an associated crossover field. The renormalization-group recursion relations are analyzed within this analytic continuation to order ε=6-d, where d is the spatial dimension. For r near unity a perturbative calculation to first order in (r-1) agrees with both the result obtained here for general r and with the approximate relation proposed by de Arcangelis et al. between the nonlinear conductivity and the noise characteristics of a linear network. For arbitrary r and d a generalization of this perturbative treatment gives (r+1)dφ(r)/dr=∂ψ(q,r)/∂q‖q=1, where φ(r) is the resistance crossover exponent and ψ(q,r) a generalized noise crossover exponent associated with ‖∂R/∂σb‖q, both quantities referred to the nonlinear system, where σb is the conductance of an individual bond. For r not near unity our results to first order in ε for φ(r) and ψ(q,r) satisfy the above relation but not that of de Arcangelis et al. For q=0, ψ(q,r)/νp is the fractal dimension of the backbone, where νp is the correlation length exponent for percolation. As is known, φ(0)/νp is an exponent associated with the chemical length, for which our result agrees with that given by Cardy and Grassberger and by Janssen.