Localization of random walks and vibrational excitations in random fractal structures

Abstract

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 r 2 ∼ r c ( N) 1−d min 〈 R(t)〉, where 〈 R(t)〉 ∼ t 1/ d w is the r.m.s. displacement of the random walker, d min is the fractal dimension of the shortest path on the fractal and r c ( N) increases logarithmically with N. For r below r 2, In 〈 P(r,t)〉 N ∼ − a p ( r〈 R(t)〉) u does not depend on N and is characterized by the exponent u = d w /( d w − 1), while for r > r 2 the coefficient a p decreases logarithmically with N and the exponent becomes u = d min d w /( d w − d min). We discuss the relevance of the results to the important problem of localization of vibrational excitations on random fractal structures.