Lévy Flights in Random Environments

We consider Lévy flights characterized by the step index $f$ in a quenched isotropic short-range random force field to one loop order. By means of a dynamic renormalization group analysis, we find that the dynamic exponent $z$ for $f<\mathrm{}2\mathrm{}$ locks onto $f$, independent of dimension and independent of the presence of weak quenched disorder. The critical dimension for $f<\mathrm{}2\mathrm{}$ is given by $d_c=2f-2\mathrm{}$. For $d<\mathrm{}{d}_c$ the disorder is relevant, corresponding to a nontrivial fixed point for the force correlation function. We also discuss the behavior of the subleading diffusive term.