Asymptotic properties of Lévy flights in quenched random fields

Long-time asymptotic behavior of the probability distribution function of Levy flights in quenched random fields is analyzed with the use of field-theoretic renormalization group. This problem has been recently studied with the aid of a dynamic renormalization group based on the momentum-shell integration method [H.C. Fogedby, Phys. Rev. E 58, 1690 (1998)]. While a great deal of the results of the quoted paper are confirmed by the present analysis, it is also shown that random field with long-range spatial correlations gives rise to asymptotic behavior with the dynamic critical exponent z less than the step index f of the Levy flights, for a finite range of values of f contrary to the conjecture that always z=f. In particular, in divergenceless random field z=d/2+1-alpha<f, when alpha<1+d/2-f and d<2+2alpha, and correlations fall off as r(-d+2alpha). The physical content of a new critical dimension proposed in the aforementioned paper in connection with the anomalous scaling of the diffusion coefficient is also discussed.