Noisy random resistor networks: renormalized field theory for the multifractal moments of the current distribution

Abstract

We study the multifractal moments of the current distribution in randomly diluted resistor networks near the percolation treshold. When an external current is applied between to terminals $x$ and $x^\prime$ of the network, the $l$th multifractal moment scales as $M_I^{(l)} (x, x^\prime) \sim | x - x^\prime |^{ψ_l /ν}$, where $ν$ is the correlation length exponent of the isotropic percolation universality class. By applying our concept of master operators [Europhys. Lett. {\bf 51}, 539 (2000)] we calculate the family of multifractal exponents $\{ψ_l \}$ for $l \geq 0$ to two-loop order. We find that our result is in good agreement with numerical data for three dimensions.