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We present a random matrix approach to study general vibrational propertiesof stable amorphous solids with translational invariance using the correlatedWishart ensemble. Within this approach, both analytical and numerical methodscan be applied. Using the random matrix theory, we found the analytical form ofthe vibrational density of states and the dynamical structure factor. Wedemonstrate the presence of the Ioffe-Regel crossover between low-frequencypropagating phonons and diffusons at higher frequencies. The reducedvibrational density of states shows the boson peak, which frequency is close tothe Ioffe-Regel crossover. We also present a simple numerical random matrixmodel with finite interaction radius, which properties rapidly converges to theanalytical results with increasing the interaction radius. For fine interactionradius, the numerical model demonstrates the presence of the quasilocalizedvibrations with a power-law low-frequency density of states.

A random matrix approach to the boson peak and Ioffe-Regel criterion in amorphous solids