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

Erdős, László ; Knowles, Antti

In: Annales Henri Poincaré, 2015, vol. 16, no. 3, p. 709-799

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    Titre
    • The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case
    Auteur
    • Erdős, László. IST Austria, Am Campus 1, 3400, Klosterneuburg, Austria
    • Knowles, Antti. Departement Mathematik, ETH Zürich, HG G 66.5, Rämistrasse 101, 8092, Zurich, Switzerland
    Type de document
    • Postprint
    Classification
    • Mathématiques
    Identifiant OAI-PMH
    • oai:doc.rero.ch:332260
    Summary
    The Altshuler-Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) 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 (Erdős and Knowles, The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case, 2013) 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 (β=1) and complex Hermitian (β=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.