Time dependent correlation functions of diffusing particles in random systems

We investigate the probability density 〈P(r, t)〉of diffusing particles in percolation systems at the percolation threshold and in self-avoiding random walks, which are considered as model systems for structural disorder. We find that 〈P(r, t)〉 is a stretched Gaussian and scales as log [if[<P(r,t)〉/〈P(r,0)〉]{reversed tilde equals}-[r/R(t)]^u, where R(t) is the root-mean-square displacement, R(t) {reversed tilde equals} t^1/dw, and u=dw/(dw-1). We also study how P(r,t) varies, for fixed distance r,and time t, for different realizations of the structural disorder. We find that the fluctuations have a broad logarithmic distribution, and the average moments 〈P^q〉 scale in a characteristic multifractal fashion.