Largest current in a random resistor network

The expected largest current in a random resistor network is shown to scale as (lnL${)}^{\alpha }$, where L is the size of the network and the exponent α depends on dimension and on the ratio of the smallest to the largest conductances in the network. This result follows from an analysis of configurations which carry large currents. The configurations which carry the largest currents in the network are shown to be funnel shaped. The proof that these configurations carry the largest currents is based upon a theorem concerning the minimum eigenvalue of a class of Sturm-Liouville problems.