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The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case

Erdős, László ; Knowles, Antti

In: Communications in Mathematical Physics, 2015, vol. 333, no. 3, p. 1365-1416

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    Title
    • The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case
    Author
    • Erdős, László. IST Austria, Am Campus 1, 3400, Klosterneuburg, Austria
    • Knowles, Antti. ETH Zürich, Zurich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Communications in Mathematical Physics, 2015, vol. 333, no. 3, p. 1365-1416. Springer Berlin Heidelberg
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:332902
    Summary
    We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson- Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it describes the correlations of the eigenvalue density in general metallic sampleswith weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev Lett 74:2563-2566, 1995). The proof is given in the current paper and its companion (Ann. H. Poincaré. arXiv:1309.5107 , 2014).