Local and average behavior in inhomogeneous superdiffusive media

TL;DR

This paper investigates the large-time behavior of random walks on inhomogeneous fractal graphs, revealing how geometric parameters influence superdiffusive behavior and highlighting differences between local and average observables in Le9vy-like processes.

Contribution

It introduces a geometric parameter b1 that governs superdiffusive behavior and analyzes the asymptotic properties of local and average observables in inhomogeneous fractal media.

Findings

Different asymptotic behaviors as a function of b1 are identified.

The root of the mean square displacement and the characteristic length may grow differently.

Average quantities are influenced by the anomalous Le9vy statistics.

Abstract

We consider a random walk on one-dimensional inhomogeneous graphs built from Cantor fractals. Our study is motivated by recent experiments that demonstrated superdiffusion of light in complex disordered materials, thereby termed L\'evy glasses. We introduce a geometric parameter $\alpha$ which plays a role analogous to the exponent characterizing the step length distribution in random systems. We study the large-time behavior of both local and average observables; for the latter case, we distinguish two different types of averages, respectively over the set of all initial sites and over the scattering sites only. The "single long jump approximation" is applied to analytically determine the different asymptotic behaviours as a function of $\alpha$ and to understand their origin. We also discuss the possibility that the root of the mean square displacement and the characteristic length of the walker distribution may grow according to different power laws; this anomalous behaviour is typical of processes characterized by L\'evy statistics and here, in particular, it is shown to influence average quantities.