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To better understand the surprising low-frequency vibrational modes instructural glasses, we study the spectra of a large ensemble of sparse randommatrices where disorder is controlled by the distribution of bond weights andnetwork coordination. We find $D(\omega)$ has three regimes: avery-low-frequency regime that can be predicted analytically using extremalstatistics, an intermediate regime with quasi-localized modes, and a plateauwith $D(\omega) \sim \omega^0$. In the special case of uniform bond weights,the intermediate regime displays $D(\omega) \sim \omega^4$, independent ofnetwork coordination and system size, just as recently discovered insimulations of structural glasses.

A simple random matrix model for the vibrational spectrum of jammed packings