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A simple model of harmonic vibrations in topologically disordered systems,such as glasses and supercooled liquids, is studied analytically by extendingEuclidean Random Matrix Theory to include vector vibrations. Rather generally,it is found that i) the dynamic structure factor shows sound-like Brillouinpeaks whose longitudinal/transverse character can only be distinguished forsmall transferred momentum, p, ii) the model presents a mechanical instabilitytransition at small densities, for which scaling laws are analyticallypredicted and confirmed numerically, iii) the Brillouin peaks persist deep intothe unstable phase, the phase transition being noticeable mostly in theirlinewidth, iv) the Brillouin linewidth scales like p square in the stablephase, and like p in the unstable one. The analytical results are checkednumerically for a simple potential. The main features of glassy vibrationspreviously deduced from scalar ERMT are not substantially altered by these newresults.Comment: 15 postscript figure

Brillouin and boson peaks in glasses from vector Euclidean random matrix theory