Critical Exponents for Diluted Resistor Networks

TL;DR

This paper uses a reformulated field theory approach to calculate the critical resistance exponent in diluted resistor networks near the percolation threshold, confirming previous results with a new interpretation of Feynman diagrams.

Contribution

It introduces an alternative diagram evaluation method for resistor networks within a reformulated field theory framework, extending the calculation of the resistance crossover exponent to second order in epsilon.

Findings

Calculated the resistance crossover exponent $$ up to second order in 

Verified previous results by Lubensky and Wang

Provided a new interpretation of Feynman diagrams for resistor networks

Abstract

An approach by Stephen is used to investigate the critical properties of randomly diluted resistor networks near the percolation threshold by means of renormalized field theory. We reformulate an existing field theory by Harris and Lubensky. By a decomposition of the principal Feynman diagrams we obtain a type of diagrams which again can be interpreted as resistor networks. This new interpretation provides for an alternative way of evaluating the Feynman diagrams for random resistor networks. We calculate the resistance crossover exponent $\phi$ up to second order in $\epsilon=6-d$, where $d$ is the spatial dimension. Our result $\phi=1+\epsilon /42 +4\epsilon^2 /3087$ verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts--model formulation of the random resistor network.