A random matrix definition of the boson peak

The density of vibrational states for glasses and jammed solids exhibits universal features, including an excess of modes above the Debye prediction known as the boson peak located at a frequency ${\omega }^{\ast }$ . We show that the eigenvector statistics for boson peak modes are universal, and develop a new definition of the boson peak based on this universality that displays the previously observed characteristic scaling ${\omega }^{\ast }\sim p^{-1\mathrm{/}2}$ . We identify a large new class of random matrices that obey a generalized global tranlational invariance constraint and demonstrate that members of this class also have a boson peak with precisely the same universal eigenvector statistics. We denote this class as boson peak random matrices, and conjecture it comprises a new universality class. We characterize the eigenvector statistics as a function of coordination number, and find that one member of this new class reproduces the scaling of ${\omega }^{\ast }$ with coordination number that is observed near the jamming transition.