Dynamical singularities in ultradiffusion

We study ultradiffusion in systems with an arbitrary distribution of energy barriers and nearest-neighbor hopping processes. As the effective temperature R is increased, we find power-law decay of the autocorrelation function with an anomalous R-dependent exponent for R$R_c$. Furthermore, we observe singular crossover at $R_c$ to normal diffusion. The value of $R_c$ depends on the average branching ratio of the system. We have also discovered a set of trees for which the R dependence of the exponent departs from the expected value based on universality arguments. Analytical results and computer experiments for several systems are presented.