ON THE LOCALIZATION OF RANDOM-WALKS AND VIBRATIONAL EXCITATIONS IN LINEAR FRACTAL STRUCTURES

We study analytically and numerically the mean probability density [P(r, t)]N of random walks and the mean vibrational amplitudes [psi(r, w)]N of localized excitations on linear random fractals, averaged over N configurations. We find that for large distances r, both functions are characterized by crossovers r(c) and r(c)', respectively, which increase logarithmically with N. For r below the respective crossover points, In[P(r,t)]N approximately -a(p)r(u) and [psi(r,omega)]N approximately -a(psi)r(gamma) do not depend on N and are characterized by the exponents u=d(w)/(d(w)-1) and gamma=1, while above the crossovers, the coefficients a(p) and a(psi) decrease logarithmically with N and the exponents become u=d(min)d(w)/(d(w)-d(min)) and gamma=d(min), where d(w) is the fractal dimension of the random walk and d(min) is the fractal dimension of the shortest path on the fractal structure. We discuss the applicability of these results to general random fractal structures.