Quantum diffusion and delocalization for band matrices with general distribution

L. Erdös, A. Knowles, Annales Henri Poincare 12 (2011) 1227–1319.

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Abstract
We consider Hermitian and symmetric random band matrices H in d ≥ dimensions. The matrix elements Hxy, indexed by x,y ∈ Λ ⊂ ℤd are independent and their variances satisfy σ2xy:= E{pipe}Hxy{pipe}2 = W-d f((x-y)/W for some probability density f. We assume that the law of each matrix element Hxy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales ≪ Wd/3. We also show that the localization length of the eigenvectors of H is larger than a factor Wd/6 times the band width W. All results are uniform in the size {pipe}Λ{pipe} of the matrix. This extends our recent result (Erdo{double acute}s and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying Σx σ2xy for all y, the largest eigenvalue of H is bounded with high probability by 2+M-2/3+e{open} for any e{open} > 0, where M:= 1/(maxx,y σ2xy).
Publishing Year
Date Published
2011-11-01
Journal Title
Annales Henri Poincare
Volume
12
Issue
7
Page
1227 - 1319
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Erdös L, Knowles A. Quantum diffusion and delocalization for band matrices with general distribution. Annales Henri Poincare. 2011;12(7):1227-1319. doi:10.1007/s00023-011-0104-5
Erdös, L., & Knowles, A. (2011). Quantum diffusion and delocalization for band matrices with general distribution. Annales Henri Poincare, 12(7), 1227–1319. https://doi.org/10.1007/s00023-011-0104-5
Erdös, László, and Antti Knowles. “Quantum Diffusion and Delocalization for Band Matrices with General Distribution.” Annales Henri Poincare 12, no. 7 (2011): 1227–1319. https://doi.org/10.1007/s00023-011-0104-5.
L. Erdös and A. Knowles, “Quantum diffusion and delocalization for band matrices with general distribution,” Annales Henri Poincare, vol. 12, no. 7, pp. 1227–1319, 2011.
Erdös L, Knowles A. 2011. Quantum diffusion and delocalization for band matrices with general distribution. Annales Henri Poincare. 12(7), 1227–1319.
Erdös, László, and Antti Knowles. “Quantum Diffusion and Delocalization for Band Matrices with General Distribution.” Annales Henri Poincare, vol. 12, no. 7, Birkhäuser, 2011, pp. 1227–319, doi:10.1007/s00023-011-0104-5.

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