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

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