Abstract
We use the recursion method to calculate the vibrational density of states\(\mathcal{N}\left( \omega \right)\) of site percolation clusters slightly above the percolation threshold. It is found that\(\mathcal{N}\left( \omega \right)\) is proportional to ω at long wavelengths. At shorter length scales,\(\mathcal{N}\left( \omega \right)\) is proportional to\(\omega ^{\bar d - 1} \), with the fraction dimension\(\bar d \approx 1.30\). The cross-over from phonon to fraction regime is characterized by a rapid rise in\(\mathcal{N}\left( \omega \right)\) in agreement with effective medium calculations.
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Lam, P.M., Bao, W. & Zheng, Z. Recursion method for the density of states and spectral dimension of percolation networks. Z. Physik B - Condensed Matter 59, 63–67 (1985). https://doi.org/10.1007/BF01325383
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DOI: https://doi.org/10.1007/BF01325383