Vibrations and random walks on random fractals: Anomalous behaviour 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 Pi, (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 ψi(r,w), for fixed r and w, show similar features. In both cases, the multifractality vanishes on deterministic fractals and on those random fractals for which the fractal dimension dx in chemical space is equal to the fractal dimension df of the structure. By relating the distribution function which represents the average over all Pi(r, t), to the averaged envelope function ψi(r, w) describing the decay of vibrational excitations, we find that the vibrational excitations (fractons) are localized with a frequency-dependent localization length λ(w); the envelope function ψ(r, w) decays exponentially for r/λ> 1. Finally we discuss the vibrational density N(w) of states for quite general random AB networks consisting of A and B bonds with force constants fAand fB, 0