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 ${\mathit{hL}}^{\mathrm{\phi }}$≪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)dα is the number of currents with α<-ln${\mathit{I}}_{\mathit{i}}$<α+dα; ${\mathit{I}}_{\mathit{i}}$ 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 ${\mathit{hL}}^{\mathrm{\phi }}$≫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.

• Received 27 May 1992

DOI:https://doi.org/10.1103/PhysRevB.46.11487