Crossover in spectral dimensionality of elastic percolation systems

Abstract

A scaling theory of the low-frequency vibrational density of states and dispersion relations for percolation systems with rotationally invariant elastic forces is presented. It is found that for the standard discrete network models, there exists a new crossover length scale ${l}_{c}$ which depends on the relative strength of the microscopic bond-stretching and bond-bending elastic force constants, such that if ${l}_{c}$>1, then (a) when the correlation length \ensuremath{\xi} is much smaller than ${l}_{c}$, the effective spectral dimension in the fracton regime is given by d\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\approxeq}(4/3), or (b) when \ensuremath{\xi} is much larger than ${l}_{c}$, there is an interesting crossover of spectral dimensionality from D\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\approxeq}0.8 to d\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\approxeq}(4/3) as frequency is increased through the fracton regime. For the random-void class of continuum percolation models, the values of these dimensions change in correspondence with the changes in the percolation elasticity exponent found earlier.