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

L. Erdös, A. Knowles, Communications in Mathematical Physics 333 (2015) 1365–1416.


Journal Article | Published | English
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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).
Publishing Year
Date Published
2015-02-01
Journal Title
Communications in Mathematical Physics
Volume
333
Issue
3
Page
1365 - 1416
IST-REx-ID

Cite this

Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 2015;333(3):1365-1416. doi:10.1007/s00220-014-2119-5
Erdös, L., & Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics, 333(3), 1365–1416. https://doi.org/10.1007/s00220-014-2119-5
Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics 333, no. 3 (2015): 1365–1416. https://doi.org/10.1007/s00220-014-2119-5.
L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case,” Communications in Mathematical Physics, vol. 333, no. 3, pp. 1365–1416, 2015.
Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 333(3), 1365–1416.
Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics, vol. 333, no. 3, Springer, 2015, pp. 1365–416, doi:10.1007/s00220-014-2119-5.

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