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~(lnL)^α, for L (the linear dimension of the network) {}ξₚ (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 Icom and scales as I_com~(lnL)^β. 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 Iₘₐₓ 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.