Transport and Scaling in Quenched 2D and 3D Lévy Quasicrystals

Abstract

We consider correlated Lévy walks on a class of two- and three-dimensional deterministic self-similar structures, with correlation between steps induced by the geometrical distribution of regions, featuring different diffusion properties. We introduce a geometric parameter $\alpha$, playing a role analogous to the exponent characterizing the step-length distribution in random systems. By a {\it single-long jump} approximation, we analytically determine the long-time asymptotic behavior of the moments of the probability distribution, as a function of $\alpha$ and of the dynamic exponent $z$ associated to the scaling length of the process. We show that our scaling analysis also applies to experimentally relevant quantities such as escape-time and transmission probabilities. Extensive numerical simulations corroborate our results which, in general, are different from those pertaining to uncorrelated Lévy-walks models.

10 pages, 11 figures; some concepts rephrased to improve on clarity; a few references added; symbols and line styles in some figures changed to improve on visibility

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• Holey Random Walks: Optics of Heterogeneous Turbid Composites
• Large-deviations for spatial diffusion of cold atoms
• Transmission probability through a Lévy glass and comparison with a Lévy walk
• Superdiffusion and Transport in 2d-systems with Lévy Like Quenched Disorder
• Scaling properties of field-induced superdiffusion in Continous Time Random Walks
• Transport in quenched disorder: light diffusion in strongly heterogeneous turbid media
• Random walks in a one-dimensional Lévy random environment
• Localization in one-dimensional chains with Lévy-type disorder
• Steering random walks with kicked ultracold atoms
• Photonic heterostructures with Levy-type disorder: statistics of coherent transmission
• Scattering lengths and universality in superdiffusive Lévy materials
• Walk dimension for light in complex disordered media
• Anomalous transmission and drifts in one-dimensional Levy structures
• Topological regulation of activation barriers on fractal substrates
• Breakdown of self-similarity in light transport