Delocalization and diffusion profile for random band matrices

L. Erdös, A. Knowles, H. Yau, J. Yin, Communications in Mathematical Physics 323 (2013) 367–416.


Journal Article | Published
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Abstract
We consider Hermitian and symmetric random band matrices H = (h xy ) in d⩾1 d ⩾ 1 dimensions. The matrix entries h xy , indexed by x,y∈(Z/LZ)d x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances sxy=E|hxy|2 s x y = E | h x y | 2 . We assume that s xy is negligible if |x − y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if W≫L4/5 W ≫ L 4 / 5 . We also show that the magnitude of the matrix entries |Gxy|2 | G x y | 2 of the resolvent G=G(z)=(H−z)−1 G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute E|Gxy|2 E | G x y | 2 . We show that, as L→∞ L → ∞ and W≫L4/5 W ≫ L 4 / 5 , the behaviour of E|Gxy|2 E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.
Publishing Year
Date Published
2013-10-01
Journal Title
Communications in Mathematical Physics
Volume
323
Issue
1
Page
367 - 416
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Cite this

Erdös L, Knowles A, Yau H, Yin J. Delocalization and diffusion profile for random band matrices. Communications in Mathematical Physics. 2013;323(1):367-416. doi:10.1007/s00220-013-1773-3
Erdös, L., Knowles, A., Yau, H., & Yin, J. (2013). Delocalization and diffusion profile for random band matrices. Communications in Mathematical Physics, 323(1), 367–416. https://doi.org/10.1007/s00220-013-1773-3
Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “Delocalization and Diffusion Profile for Random Band Matrices.” Communications in Mathematical Physics 323, no. 1 (2013): 367–416. https://doi.org/10.1007/s00220-013-1773-3.
L. Erdös, A. Knowles, H. Yau, and J. Yin, “Delocalization and diffusion profile for random band matrices,” Communications in Mathematical Physics, vol. 323, no. 1, pp. 367–416, 2013.
Erdös L, Knowles A, Yau H, Yin J. 2013. Delocalization and diffusion profile for random band matrices. Communications in Mathematical Physics. 323(1), 367–416.
Erdös, László, et al. “Delocalization and Diffusion Profile for Random Band Matrices.” Communications in Mathematical Physics, vol. 323, no. 1, Springer, 2013, pp. 367–416, doi:10.1007/s00220-013-1773-3.

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