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∼$I^r$. 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${\Vert }_q$=1, where φ(r) is the resistance crossover exponent and ψ(q,r) a generalized noise crossover exponent associated with ‖∂R/∂${\sigma }_b$ ${\Vert }^q$, both quantities referred to the nonlinear system, where ${\sigma }_b$ 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)/${\nu }_p$ is the fractal dimension of the backbone, where ${\nu }_p$ is the correlation length exponent for percolation. As is known, φ(0)/${\nu }_p$ is an exponent associated with the chemical length, for which our result agrees with that given by Cardy and Grassberger and by Janssen.