VIBRATIONS AND RANDOM-WALKS ON RANDOM FRACTALS - ANOMALOUS BEHAVIOR AND MULTIFRACTALITY

We discuss vibrational properties and random walks on random fractal structures, in particular on the infinite percolation cluster at criticality. We show that the probabilities P(i)(r, t) of finding a random walker after t time steps on a site i at distance r from its starting point are characterized by a logarithmically broad distribution and display multifractal features. The corresponding vibrational amplitudes psi(i)(r, omega), for fixed r and omega, show similar features. In both cases, the multifractality vanishes on deterministic fractals and on those random fractals for which the fractal dimension d(l) in chemical space is equal to the fractal dimension d(f) of the structure. By relating the distribution function <P(r, t)> which represents the average over all P(i)(r, t), to the averaged envelope function psi(r, omega) describing the decay of vibrational excitations, we find that the vibrational excitations (fractons) are localized with a frequency-dependent localization length lambda(omega); the envelope function psi(r, omega) decays exponentially for r-lambda much greater than 1. Finally we discuss the vibrational density N(omega) of states for quite general random AB networks consisting of A and B bonds with force constants f(A) and f(B), 0 < f(B) much less than f(A) and develop a scaling theory for N(omega) near criticality.