The Altshuler–Shklovskii formulas for random band matrices II: The general case

L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.


Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2015-03-01
Journal Title
Annales Henri Poincare
Volume
16
Issue
3
Page
709 - 799
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Cite this

Erdös L, Knowles A. The Altshuler–Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 2015;16(3):709-799. doi:10.1007/s00023-014-0333-5
Erdös, L., & Knowles, A. (2015). The Altshuler–Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare, 16(3), 709–799. https://doi.org/10.1007/s00023-014-0333-5
Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random Band Matrices II: The General Case.” Annales Henri Poincare 16, no. 3 (2015): 709–99. https://doi.org/10.1007/s00023-014-0333-5.
L. Erdös and A. Knowles, “The Altshuler–Shklovskii formulas for random band matrices II: The general case,” Annales Henri Poincare, vol. 16, no. 3, pp. 709–799, 2015.
Erdös L, Knowles A. 2015. The Altshuler–Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 16(3), 709–799.
Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random Band Matrices II: The General Case.” Annales Henri Poincare, vol. 16, no. 3, Springer, 2015, pp. 709–99, doi:10.1007/s00023-014-0333-5.

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