Localization of random walks and vibrational excitations in random fractal structures

We study analytically and numerically the mean probability density P(r, t) N of random walks on random fractals, averaged over N configurations. We find that for large distances r, P(r, t) N is characterized by a crossover at r r2 rc(N)1−dmin R(t) , where R(t) t1/dw is the r.m.s. displacement of the random walker, dmin is the fractal dimension of the shortest path on the fractal and rc(N) increases logarithmically with N. For r below r2, In P(r,t) N −ap(r R(t) )u does not depend onN and is characterized by the exponent u = dw/(dw − 1), while for r > r2 the coefficient ap decreases logarithmically with N and the exponent becomes u = dmindw/(dw − dmin). We discuss the relevance of the results to the important problem of localization of vibrational excitations on random fractal structures