Lévy flights in a quenched jump length field: a real space renormalization group approach

Abstract

We study the d-dimensional walk of a particle in the presence of a quenched random field of jump lengths l( x ) drawn from an algebraic distribution p( l)∼ l −1− f with the Lévy exponent f. A real space renormalization procedure shows that the upper critical dimension is given by d c = f. Furthermore, the renormalization group approach indicates, that the dynamic z exponent characterizing the walk of a particle in such a field is changed from the annealed value z= f (usual Lévy flight) to z= f+2( f− d)/ d (quenched case). This change of the dynamical exponent z= f for an annealed Lévy walker into the exponent z> f for the same Lévy walker in a quenched jump length field is a remarkable result: an intrinsic anomalous diffusion is changed into another anomalous diffusion by an external disordered jump length field.