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

Abstract

The recent realization of a "Levy glass" (a three-dimensional optical material with a Levy distribution of scattering lengths) has motivated us to analyze its one-dimensional analogue: A linear chain of barriers with independent spacings s that are Levy distributed: p(s)~1/s^(1+alpha) for s to infinity. The average spacing diverges for 0<alpha<1. A random walk along such a sparse chain is not a Levy 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^(-alpha) ln L for 0<alpha<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.

arXiv comment5 pages, 2 figures; introduction expanded, references added

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• One-Dimensional Stochastic Lévy-Lorentz Gas
• Semiclassical theory of shot-noise suppression
• Electronic shot noise in fractal conductors
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• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
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• Rare Events in Extreme Value Statistics of Jump Processes with Power Tails
• Statistics of coherent waves inside media with Lévy disorder