Log-periodic modulation in one-dimensional random walks

We have studied the diffusion of a single particle on a one-dimensional lattice. It is shown that, for a self-similar distribution of hopping rates, the time dependence of the mean-square displacement follows an anomalous power law modulated by logarithmic periodic oscillations. The origin of this modulation is due to the dependence of the diffusion coefficient on the length scale. Both the random walk exponent and the period of the modulation are analytically calculated and confirmed by Monte Carlo simulations.