Vibrational excitations in percolation: Localization and multifractality

We discuss localized excitations on the incipient infinite percolation cluster. Assuming a simple exponential decay of the amplitudes ${\mathrm{\psi }}_{\mathit{i}}$ in terms of the chemical (minimal) path, we show theoretically that the ψ’s are characterized by a logarithmically broad distribution, and display multifractal features as a function of the Euclidean distance. The moments of ${\mathrm{\psi }}_{\mathit{i}}$ exhibit novel crossover phenomena. Our numerical simulations of fractons exhibit a nontrivial distribution of localization lengths, even when the chemical distance is fixed. These results are explained via a generalization of the theory.