On the localization of random walks and vibrational excitations in linear fractal structures

We study analytically and numerically the mean probability density N of random walks and the mean vibrational amplitudes 〈ψ(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 rc and r′c, respectively, which increase logarithmically with N. For r below the respective crossover points, In N ≈ -aPru and <ψ(r, w) >N ≈ -aψrγ do not depend on N and are characterized by the exponents u = dw/(dw - 1) and γ = 1, while above the crossover, the coefficients ap and aψ decrease logarithmically with N and the exponents become u = dmindw/(dw - dmin) and γ = dmin, where dw is the fractal dimension of the random walk and dmin is the fractal dimension of the shortest path on the fractal structure. We discuss the applicability of these results to general random fractal structures.