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 ξ is much smaller than $l_c$, the effective spectral dimension in the fracton regime is given by d̃≊(4/3), or (b) when ξ is much larger than $l_c$, there is an interesting crossover of spectral dimensionality from D̃≊0.8 to d̃≊(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.