Microscopic dynamical exponents for random-random directed walk on a one-dimensional lattice with quenched disorder

Abstract

We demonstrate that the dynamical exponent for the time dependence of the coordinate, previously found for an average over disorder, is already present in any realization of a given sample. This ergodicity comes from the existence of a scaling law for the probability distribution of the parameter defining the asymptotic dynamical regime. The self-averaging or non-self-averaging properties of the normal or anomalous phases are direct consequences of this result.