# Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

Erdös L, Schlein B, Yau H. 2009. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Annals of Probability. 37(3), 815–852.

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Author

Erdös, László

^{ISTA}^{}; Schlein, Benjamin; Yau, Horng-TzerAbstract

We consider N×N Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. We study the connection between eigenvalue statistics on microscopic energy scales η≪1 and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order η∼log N/N. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales η≫N−2/3. We show that most eigenvectors are fully delocalized in the sense that their ℓp-norms are comparable with N1/p−1/2 for p≥2, and we obtain the weaker bound N2/3(1/p−1/2) for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.

Publishing Year

Date Published

2009-01-01

Journal Title

Annals of Probability

Volume

37

Issue

3

Page

815 - 852

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### Cite this

Erdös L, Schlein B, Yau H. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices.

*Annals of Probability*. 2009;37(3):815-852. doi:10.1214/08-AOP421Erdös, L., Schlein, B., & Yau, H. (2009). Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices.

*Annals of Probability*. Institute of Mathematical Statistics. https://doi.org/10.1214/08-AOP421Erdös, László, Benjamin Schlein, and Horng Yau. “Semicircle Law on Short Scales and Delocalization of Eigenvectors for Wigner Random Matrices.”

*Annals of Probability*. Institute of Mathematical Statistics, 2009. https://doi.org/10.1214/08-AOP421.L. Erdös, B. Schlein, and H. Yau, “Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices,”

*Annals of Probability*, vol. 37, no. 3. Institute of Mathematical Statistics, pp. 815–852, 2009.Erdös, László, et al. “Semicircle Law on Short Scales and Delocalization of Eigenvectors for Wigner Random Matrices.”

*Annals of Probability*, vol. 37, no. 3, Institute of Mathematical Statistics, 2009, pp. 815–52, doi:10.1214/08-AOP421.