Euclidean random matrix theory: low-frequency non-analyticities and Rayleigh scattering

Abstract

By calculating all terms of the high-density expansion of Euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system, we show that the low-frequency behavior of the self-energy is given by Σ(k, z) ∝ k 2 z d/2 and not Σ(k, z) ∝ k 2 z (d−2)/2, as claimed previously. This implies the presence of Rayleigh scattering and long-time tails of the velocity autocorrelation function of the analogous diffusion problem of the form Z(t) ∝ t (d+2)/2.