Abstract
The asymmetric diffusion of a particle in a random one-dimensional medium can be described by a model of random potential with positive spectrum closely linked to supersymmetric quantum mechanics. We obtain analytical expressions for the density of states ρ(ε) (inverse relaxation time spectrum). This allows us to compute the averaged probability of return at any time. At zero energy ρ(ε) exhibits a variety of singular behaviours with a continuously varying exponent. This corresponds to the different phases of the diffusion problem at large time, including Sinaï's behaviour ⟨x2(t)⟩ = C ln4t. The validity of the dynamical-scaling assumption is discussed.