Transport and scaling in quenched two- and three-dimensional Lévy quasicrystals

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 α, playing a role analogous to the exponent characterizing the step-length distribution in random systems. By a single-long-jump approximation, we analytically determine the long-time asymptotic behavior of the moments of the probability distribution as a function of α and of the dynamic exponent z associated with 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-walk models.