On the propagator of Sierpinski gaskets

The authors present numerical calculations for the propagator P(r, t), the probability to reach a distance r at time t having started at the origin at t=0, on Sierpinski gaskets. The results are confronted with approximate analytical expressions. It is shown that P(r, t) approximately t-ds/2 Pi ( xi ), where xi is the scaling variable: xi =r/t1/dw. In the short- xi regime the scaling function follows the form Pi ( xi ) approximately exp(-c1 xi dw), while for large xi , Pi ( xi ) is given asymptotically by Pi ( xi ) approximately xi alpha exp(-c2 xi nu ), with alpha =(df-dw/2)/(dw-1) and nu =dw/(dw-1). This result extends the previously derived expressions. The numerically observed oscillations which are superimposed on the power-law decay of the autocorrelation function P(r=0, t) are analysed in terms of typical residence times on hierarchical substructures.