Short-time regime propagator in fractals

The propagator P(r,t) for fractals in the short-time regime, i.e., the probability of finding at distance r at time t a particle that diffuses in a fractal substrate when xi = r/root 2Dt(1/dw) much greater than 1, is studied in order to elucidate its full functional form. For finitely ramified fractals it is shown land, for any other self-similar media, conjectured) that the short-time propagator is given by P(r,t)approximate to P(0)t(-ds/2)xi(alpha)exp(-c xi(nu) where nu = d(w)/(d(w)-1) and alpha = nu/2-d(f), d(f) and d(w) being the fractal and random walk dimension of the medium, respectively. The value for nu agrees with that generally accepted. However, our result for the as yet not well established value of alpha differs from other recent proposals. We have checked these various short-time propagator proposals by comparing them to the short-time propagator calculated numerically for the Sierpinsky gasket. Our numerical results are precise enough to clearly support the validity of the short-time propagator proposed here (in particular, the validity of our relation for alpha) and to rule out the others.