Size and location of the largest current in a random resistor network

The largest current in the bonds of a random resistor network (RRN) is shown to have an anomalous size dependence given by $I_{\max \mathrm{\sim }(\mathrm{l}\mathrm{n}\mathit{L}{)}^{\alpha }}$, for L (the linear dimension of the network) ≫${\xi }_p$ (the percolation correlation length). A second important current scale is the one that leads to the eventual failure of the RRN when it is considered to be a network of fuses. This second current is defined to be $I_{\mathrm{com}}$ and scales as $I_{\mathrm{com}\mathrm{\sim }(\mathrm{l}\mathrm{n}\mathit{L}{)}^{\beta }}$. Analytic arguments are presented to support the inequality 1/[2(D-1)]≲β≤α≲1, where D is the spatial dimension. Numerical simulations in two dimensions support this, and in addition show that the bond carrying $I_{\max }$ is often near the free surfaces of the RRN. This statement is quantified by the ratio of surface to bulk probabilities, and this ratio is shown to increase algebraically with exponent x=0.30±0.05 in two dimensions.