Absence of the hump in the density of states of percolating clusters

The vibrational densities of states of two-dimensional percolating clusters with very large cluster size are investigated using a novel numerical method. It is confirmed that the fracton density of states is proportional to ${\omega }^{\frac{1}{3}}$ in accord with the conjecture of Alexander and Orbach. We find that there is no notable steepness or hump in the phonon-fracton crossover region, in conflict with the prediction of the effective-medium theory. It is shown that the fracton modes are localized in "peninsulas" in percolating clusters.