Annales Henri Poincaré

, Volume 16, Issue 3, pp 709–799 | Cite as

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

  • László Erdős
  • Antti KnowlesEmail author


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.


Density Correlation Band Matrix Band Matrice Versus Main Random Band 
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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.IST AustriaKlosterneuburgAustria
  2. 2.Departement Mathematik, ETH ZürichHG G 66.5ZurichSwitzerland

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