L\'{e}vy flights in random environments

TL;DR

This paper analyzes Lévy flights in random environments, revealing that the dynamic exponent equals the step index for f<2 and identifying the critical dimension where disorder becomes relevant.

Contribution

It provides a dynamic renormalization group analysis showing the invariance of the dynamic exponent and the dependence of the critical dimension on the step index.

Findings

Dynamic exponent z equals step index f for f<2.

Critical dimension depends on step index: d_c=2f-2.

Disorder is relevant below the critical dimension.

Abstract

We consider L\'{e}vy flights characterized by the step index $f$ in a quenched random force field. By means of a dynamic renormalization group analysis we find that the dynamic exponent $z$ for $f<2$ locks onto $f$, independent of dimension and {\em independent} of the presence of weak quenched disorder. The critical dimension, however, depends on the step index $f$ for $f<2$ and is given by $d_c=2f-2$. For $d<d_c$ the disorder is {\em relevant}, corresponding to a non trivial fixed point for the force correlation function.