The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case Journal Article


Author(s): Erdős, László; Knowles, Antti
Article Title: The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case
Affiliation IST Austria
Abstract: 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).
Journal Title: Communications in Mathematical Physics
Volume: 333
Issue 3
ISSN: 1432-0916
Publisher: Springer  
Date Published: 2015-02-01
Start Page: 1365
End Page: 1416
Sponsor: Partially supported by SFB-TR 12 Grant of the German Research Council.
URL:
DOI: 10.1007/s00220-014-2119-5
Open access: yes (repository)