Local semicircle law and complete delocalization for Wigner random matrices

L. Erdös, B. Schlein, H. Yau, Communications in Mathematical Physics 287 (2009) 641–655.

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Abstract
We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales n ≫ N -1 (log N) 8 . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the -tempℓ∞-norm of the corresponding eigenvectors is of order O(N -1/2), modulo logarithmic corrections. The upper bound O(N -1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.
Publishing Year
Date Published
2009-04-01
Journal Title
Communications in Mathematical Physics
Volume
287
Issue
2
Page
641 - 655
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Erdös L, Schlein B, Yau H. Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics. 2009;287(2):641-655. doi:10.1007/s00220-008-0636-9
Erdös, L., Schlein, B., & Yau, H. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics, 287(2), 641–655. https://doi.org/10.1007/s00220-008-0636-9
Erdös, László, Benjamin Schlein, and Horng Yau. “Local Semicircle Law and Complete Delocalization for Wigner Random Matrices.” Communications in Mathematical Physics 287, no. 2 (2009): 641–55. https://doi.org/10.1007/s00220-008-0636-9.
L. Erdös, B. Schlein, and H. Yau, “Local semicircle law and complete delocalization for Wigner random matrices,” Communications in Mathematical Physics, vol. 287, no. 2, pp. 641–655, 2009.
Erdös L, Schlein B, Yau H. 2009. Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics. 287(2), 641–655.
Erdös, László, et al. “Local Semicircle Law and Complete Delocalization for Wigner Random Matrices.” Communications in Mathematical Physics, vol. 287, no. 2, Springer, 2009, pp. 641–55, doi:10.1007/s00220-008-0636-9.

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