Current distribution in the two-component hierarchical percolation model

We present an analytic solution of the current distribution in the hierarchical diamond lattice, which consists of two types of conductance, parametrized by the ratio h of poor to good conductance. Because of the iterative nature of the model, we are able to find exact recursion relations for the current distribution. For an extremely small h such that hL^φ{}1, where L is the size of the lattice and {φ} the crossover exponent, we find that D({α}){}exp{-A[{α}-B(lnL)²]/(lnL)³}, where D({α})d{α} is the number of currents with {α}-lnIᵢ+d{α}; Iᵢ is the current in bond i and A,B are constants. The current distribution is well approximated by a Gaussian with the mean varying with size as (lnL)² while the variance varies as (lnL)³. As a result, both the most probable current and the minimum current scale in the same way with L as exp[-const(lnL)²]. In the opposite limit hL^φ{}1, the distribution reduces to the trivial form of the one-component lattice. We examine the current distribution in a two-component random resistor network in the light of the analytic results.