Diffusion on fractals with singular waiting-time distribution

We study diffusion in lattices of arbitrary dimensions with a power-law distribution of waiting times, P±-2, ±<11. Using general scaling arguments we find that the asymptotic behavior of the mean-square displacement of a random walker is given &, where d»w=dw for ±<0 and d»w=dw{1+ds±/[2(1-±)]} for 0±<1 and ds2. Here dw is the (conventional) diffusion exponent for constant waiting times and ds is the fracton dimension of the substrate. Our expression for d»w is general and holds for Euclidean lattices as well as for random and deterministic fractals. We have also investigated scaling properties of the distribution function PI(l,t) and the corresponding moments lq, where l is the chemical distance the walker traveled in time t. To test our theoretical expressions we have performed extensive computer simulations on the incipient percolation cluster in d=2, using the exact enumeration method. The numerical results agree well with the theoretical predictions.