Multifractal features of random walks and localized vibrational excitations on random fractals: Dependence on the averaging procedure

We show explicitly that the range of multifractality of random walks and localized vibrational excitations on random fractals depends crucially on the number N-av of configurations involved in the averaging procedure. By studying analytically and numerically the moments [P-q(r, t)](Nav) of the probability density of random walks on random walk trails and percolation clusters at criticality, averaged over N-av configurations, we find that for fixed distance r and time t multifractality occurs only for q values below a critical value q,(N,,) that increases logarithmically with N-av. We predict similar features for the moments of the amplitudes [Psi(q)(r, omega)](Nav) of localized vibrational excitations.