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)d{φ}(r)/dr={∂}{ψ}(q,r)/{∂}qq=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 ${{σ}}b$ 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.