The Spectra of Vibrational Modes of Random Elastic Networks
Marija Vucelja, Rockefeller University


The mechanical, dynamical and thermodynamical properties of amorphous solids are far less understood than those of crystalline solids. The analysis of these systems is complicated due to the presence of quenched disorder and vastly different interaction strengths between the constituents of these materials. Here we focus on the elastic properties of such materials. We look at spectral properties of random elastic networks which provide a good toy-model of disordered solids. We also study ensembles of random Laplacian matrices. Using the Cavity method, a sort of Bethe-Peierls iterative method, in the limit of small heterogeneities of the connectivity, we derive approximate analytical expressions for the spectral densities of these ensembles. In the Laplacian case the analytics and numerics match remarkably well at all connectivities. Our approximation also works reasonably well for the non-singular part of the vibrational spectra of random elastic networks. The present treatment is a promising starting point for an analytical description of macroscopic properties of such solids.