Transport and Scaling in Quenched 2D and 3D L\'evy Quasicrystals

TL;DR

This paper analyzes correlated Le9vy walks on self-similar 2D and 3D structures, deriving asymptotic behaviors of moments and applying results to experimental quantities, with numerical validation.

Contribution

It introduces a geometric parameter lpha that characterizes correlated Le9vy walks on fractal structures and analytically determines their long-time asymptotic behavior.

Findings

Scaling analysis applies to escape-time and transmission probabilities.

Results differ from uncorrelated Le9vy-walk models.

Numerical simulations confirm analytical predictions.

Abstract

We consider correlated L\'evy 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 L\'evy-walks models.