Minimum current in the two-component random resistor network

We have found from numerical simulation that the minimum current in the two-component random resistor network at criticality scales anomalously with the ratio h of poor to good conductances: ${\mathit{I}}_{\min }$(h)≊exp[-const×(lnh${)}^2$]. Exact analytic calculations in the diamond fractal confirm the result. In addition, we obtain a crossover behavior ${\mathit{I}}_{\min }$(h)/${\mathit{I}}_{\min }$(1)=exp[-const×(lnh${)}^2$]H(${\mathit{hL}}^{\mathrm{\phi }}$), where L is the size of the network, H is a function describing the crossover from fractal to homogeneous behaviors, and φ is the crossover exponent. The exponential prefactor is analogous to the behavior of left-sided multifractality in diffusion-limited aggregations.