Universal Collective Behavior of the Vibration Spectrum of Highly Connected Disordered Systems

We study small oscillations of highly connected systems, which represent a limit opposite to the more familiar case of disordered crystals. As a concrete example we analyze the vibrational spectra of composite pendula. Remarkably, these spectra exhibit universal behavior with non-vanishing zero-frequency limit of the density of phonon states. This universality is captured by a natural random matrix model for such systems. We analyze this model using S-transforms of free probability theory and obtain the density of modes explicitly, in the limit of large system size.