Dynamics of percolating networks

The dynamic properties of the 2D percolation network are investigated, using the spectral moments method. The density of states of percolating clusters of size 106 or larger is calculated at the percolation threshold pc and at different occupation probabilities varying from pc to p=1. As p increases, the phonon, fracton and high frequency regimes appear. These results can be interpreted by assuming that as p increases, the percolating cluster becomes homogeneous, giving rise to a phonon regime at low frequencies and to an accumulation of modes at the Van Hove singularities of the square lattice. The exponent of the coherence length away from the critical point is found to be identical with the mean field value. The localization of modes has been studied by computing the Green functions. Localized and extended modes are present in all the spectrum at pc, except in the high-frequency zone where all modes are strongly localized.