Vibrational Spectrum of Granular Packings with Random Matrices

It has previously been proposed that the vibrational spectrum of a jammed solid can be approximately obtained via random matrix theory. We observe that random matrix theory cannot explain the mean density of states but it should be able to predict universal properties of the spectrum including the correlations of the density of states. Consistent with this expectation we demonstrate good agreement between dynamical numerical simulations of granular bead packs and the analytic predictions of the Laguerre orthogonal ensemble of random matrix theory. At the same time there is clear disagreement with the predictions of the Gaussian orthogonal ensemble which establishes the Laguerre ensemble as the correct random matrix description of the jammed vibrational spectrum. We also construct a random lattice model which is a physically motivated variant on the random matrix theory. Numerical calculations reveal that this model is able to explain the mean density of states while also retaining the correct correlations obtained from the Laguerre orthogonal ensemble. We propose that the random lattice model can therefore be applied to understand not only the spectrum but more general properties of bead packs including the spatial structure of modes both at the jamming point and far from it.