εExpansion for the Conductivity of a Random Resistor Network

Abstract

We present a reanalysis of the renormalization-group calculation to first order in $\epsilon =6-d$, where $d$ is the spatial dimensionality, of the exponent, $t$, which describes the behavior of the conductivity of a percolating network at the percolation threshold. If we set $t=(d-2\mathrm{})\mathrm{}{\nu }_p+\zeta$, where ${\nu }_p$ is the correlation-length exponent, then our result is $\zeta =1+\left(\frac{\epsilon }{42}\right)$. This result clarifies several previously paradoxical results concerning resistor networks and shows that the Alexander-Orbach relation breaks down at order $\epsilon$.