Noise exponents of the random resistor network

Abstract

We consider the critical properties of the two-point resistance and its fluctuations due to microscopic noise in a randomly diluted resistor network near the percolation threshold $p_c$. We introduce a n×m replicated Hamiltonian in order to treat separately the configuration average over the randomly occupied bonds denoted [ ${]}_{\mathrm{av}}$ and the average over probability distribution function of the fluctuating microscopic bond conductance, denoted { ${\mathrm{\}}}_f$. We evaluate a family of exponents {${\psi }_l$} (l=2,3,. . .) whose values are 1+O(ε) with ε=6-d where d is the spatial dimensionality. Each ${\psi }_l$ governs the critical behavior of the lth cumulant of the resistance between the sites x,x’ conditionally averaged subject to the sites being in the same cluster such that $C_R$¯ ${}_{}{}^{\left(l\right)}{}_{}{}^{}$(x,x’)∼‖x- & for p near $p_c$, where ${\nu }_p$ is the correlation-length exponent for percolation. Furthermore, ${\psi }_2$=1+ε/105 determines the dependence of the variance of the resistance in a finite network on size L as &. .AE