Localization in one-dimensional random random walks

Diffusion in a one-dimensional random force field leads to interesting localization effects, which we study using the equivalence with a directed walk model with traps. We show that although the average dispersion of positions <([x(2)] - [x](2))over bar> diverges for long times, the probability that two independent particles occupy the same site tends to a finite constant in the small bias phase of the model. Interestingly, the long-time properties of this off-equilibrium, ageing phase is similar to the equilibrium phase of the random energy model.