Random matrix theory approach to vibrations near the jamming transition

It has been shown that the dynamical matrix $M$ describing harmonic oscillations in granular media can be represented in the form $M=AA^{\mathrm{T}}$, where the rows of the matrix $A$ correspond to the degrees of freedom of individual granules and its columns correspond to elastic contacts between granules. Such a representation of the dynamical matrix makes it possible to estimate the density of vibrational states with the use of the random matrix theory. The found density of vibrational states is approximately constant in a wide frequency range $\omega_-<\omega<\omega_+$, which is determined by the ratio of the number of degrees of freedom to the total number of contacts in the system, which is in good agreement with the results of the numerical experiments.