The Altshuler-Shklovskii Formulas for Random Band Matrices II: the General Case

Abstract

The Altshuler-Shklovskii formulas [1] predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion [2], we prove these formulas for random band matrices. In [2] we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler-Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric $(\beta=1)$ and complex Hermitian $(\beta=2)$ models, and track the transition for the mesoscopic density-density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler-Shklovskii formulas.