Localization of Fractons, Random Walks and Linear Polymers in Percolation

We review analytical and numerical results for the vibrational amplitudes of localized excitations, the probability distribution of random walks and the distribution of linear polymers (modeled by self-avoiding walks of N steps) on percolation structures at criticality. Our numerical results show that the fluctuations of these quantities, at fixed shortest-path distance ("chemical length") ℓ from the center of localization, are considerably smaller than at fixed Euclidean distance r from the center. Using this fact, we derive via convolutional integrals explicit expressions for the averaged functions in r-space, and show analytically and numerically that three different localization regimes occur. In the short-distance regime, remarkably, the averages show a universal spatial decay behavior, with the same exponent for both fractons and random walks, while in the asymptotic regime, the averages depend explicitly on the number of configurations considered.