## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

4-1-1987

#### Publication Source

Physical Review B

#### Volume

35

#### Issue

10

#### Start Page

5056

#### Last Page

5065

#### DOI

10.1103/PhysRevB.35.5056

#### Abstract

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)*d*φ(*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/∂σ

_{b}‖

^{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*)/ν

_{p}is the fractal dimension of the backbone, where ν

_{p}is the correlation length exponent for percolation. As is known, φ(0)/ν

_{p}is an exponent associated with the chemical length, for which our result agrees with that given by Cardy and Grassberger and by Janssen.

#### Recommended Citation

Harris, A.
(1987).
Field-Theoretic Formulation of the Randomly Diluted Nonlinear Resistor Network.
*Physical Review B,*
*35*
(10),
5056-5065.
http://dx.doi.org/10.1103/PhysRevB.35.5056

**Date Posted:** 12 August 2015

This document has been peer reviewed.