Nonalgebraic length dependence of transmission through a chain of barriers with a Lévy spacing distribution

The recent realization of a “Lévy glass” (a three-dimensional optical material with a Lévy distribution of scattering lengths) has motivated us to analyze its one-dimensional analog: A linear chain of barriers with independent spacings s that are Lévy distributed: p(s)∝s^−1−α for s→∞. The average spacing diverges for 0<α<1. A random walk along such a sparse chain is not a Lévy walk because of the strong correlations of subsequent step sizes. We calculate all moments of conductance (or transmission), in the regime of incoherent sequential tunneling through the barriers. The average transmission from one barrier to a point at a distance L scales as L^−α ln L for 0<α<1. The corresponding electronic shot noise has a Fano factor (∝ average noise power/average conductance) that approaches 1/3 very slowly, with 1/ln L corrections.