LEVY FLIGHTS IN QUENCHED RANDOM FORCE-FIELDS

Levy flights, characterized by the microscopic step index f, are for f <2 (the case of rare events) considered in short-range and long-range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f-2 in the sh ort-range case and the critical fall-off exponent 2f-2 in the long-ran ge case. By means of a dynamic renormalization-group analysis based on the momentum shell integration method, we determine flows, fixed poin t, and the associated scaling properties for the probability distribut ion and the frequency and wave number dependent diffusion coefficient. Unlike the case of ordinary Brownian motion in a quenched force field characterized by a single critical dimension or fall-off exponent d=2 , two critical dimensions appear in the Levy case. A critical dimensio n (or fall-off exponent) d=f below which the diffusion coefficient exh ibits anomalous scaling behavior, i.e., algebraic spatial behavior and long time tails, and a critical dimension (or fall-off exponent) d=2f -2 below which the farce correlations characterized by a nontrivial fi xed point become relevant. As a general result we find in all cases th at the dynamic exponent z, characterizing the mean square displacement , locks onto the Levy index f, independent of dimension and independen t of the presence of weak quenched disorder.