Self-consistent Euclidean-random-matrix theory

In the present survey we address the vibrational properties of a disordered mass-spring model, in which the spring constants depend exponentially on the distance between the mass positions ('Euclidean-random-matrix model', ERM). Starting from the high-density expansion for this model, introduced by Giorgio Parisi and his coworkers, we present a self-consistent approximation for the vibrational spectrum (SCERM) derived by two of the authors. By a further simplification we arrive at an ERM version of the self-consistent Born approximation (ERM-SCBA). The two approximation schemes describe correctly the transition to a Debye spectrum at low frequencies. In this regime Rayleigh scattering is predicted, which is shown to be a general feature of ERM-type models. Technically Rayleigh scattering involves a non-analyticity of the self energy, which, for the mathematically equivalent transport model, leads to a long-time tail of the velocity autocorrelation function. In the vicinity of an instability the theory predicts both the occurence of a boson peak and anomalous sound attenuation. Finally, we discuss briefly the low-density regime, which is governed by percolation physics. (paper)