## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

5-1-1987

#### Publication Source

Physical Review B

#### Volume

35

#### Issue

13

#### Start Page

6987

#### Last Page

6996

#### DOI

10.1103/PhysRevB.35.6987

#### Abstract

The randomly diluted resistor network is formulated in terms of an *n*-replicated *s*-state Potts model with a spin-spin coupling constant *J* in the limit when first *n*, then *s*, and finally *1/J* go to zero. This limit is discussed and to leading order in *1/J* the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the *xy* model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in *ε=6-d*, where *d* is the spatial dimensionality. The recursion relations are shown to be the same as for the *xy* model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as φ_{1}=1+ε/42, and determines the critical exponent, *t* for the conductivity of the randomly diluted resistor network at concentrations, *p*, just above the percolation threshold: *t*=(*d*-2)*ν*+φ_{1}, where *ν* is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving *φ=1* to all orders in *ε*. The new result for φ_{1} removes the paradox associated with the numerical result that *t>1* for *d=2*, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the *ε* expansion.

#### Recommended Citation

Harris, A.,
&
Lubensky, T. C.
(1987).
Potts-Model Formulation of the Random Resistor Network.
*Physical Review B,*
*35*
(13),
6987-6996.
http://dx.doi.org/10.1103/PhysRevB.35.6987

**Date Posted:** 12 August 2015

This document has been peer reviewed.