Localization af random walks and vibrational excitations in random fractal struct

We study analytically and numerically the mean probability density (P(r,t))~ of random walks on random fractals, averaged over N configurations. We find that for large distances r’, (P(r,t))~ is characterized by a crossover at r 5 TV - ~=(~)*-~min(~~~}) , where (R(t)) - tlfdw is the r.m.s. displacement of the random walker, dmb is the fractal dimension of the shortest path on the fractal and r,(N) increases logarithmically with N. For r below rs, In(P(r, tf)~ - -ap (r/(R(t)f)” does not depend on iV and is characterized by the exponent ?I = &/(& - I), while for T > ‘2 the coefficient cbp decreases 1ogarith~~alIy with N and the exponent becomes g = ~~~~/(~~ - d,.&. We discuss the relevance of the results to the important problem of localization of vibrational excitations on random fractal structures.