Diffusion On A One-dimensional Structure With Hierarchical Waiting-time Distribution

The problem of particle diffusion on a one-dimensional structure with a hierarchical distribution of waiting times is analyzed by an exact renormalization-group method. A detailed discussion is given of the scaling of several quantities like the mean displacement, the autocorrelation function, and the probability of finding the particle in a subset of sites, all having the same waiting time. A transition from anomalous-to-normal dynamical regimes occurs upon variation of the parameters specifying the problem. Interesting issues like the effect of reflecting and absorbing boundaries, or the possibility of logarithmic corrections for the scaling behavior, are analyzed for the first time in the context of models of ultrametric dynamics. Some of the scaling predictions are confirmed numerically. The results are of methodological interest in view of further possible applications of renormalization-group strategies to related problems.