The Altshuler–Shklovskii formulas for random band matrices II: The general case
Erdös, László
Knowles, Antti
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.
Springer
2015
info:eu-repo/semantics/article
doc-type:article
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http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/1864
Erdös L, Knowles A. The Altshuler–Shklovskii formulas for random band matrices II: The general case. <i>Annales Henri Poincare</i>. 2015;16(3):709-799. doi:<a href="https://doi.org/10.1007/s00023-014-0333-5">10.1007/s00023-014-0333-5</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00023-014-0333-5
info:eu-repo/grantAgreement/EC/FP7/338804
info:eu-repo/semantics/openAccess