# Wegner estimate and level repulsion for Wigner random matrices

Erdös L, Schlein B, Yau H. 2010. Wegner estimate and level repulsion for Wigner random matrices. International Mathematics Research Notices. (3), 436–479.

Journal Article | Published
Author
Erdös, LászlóIST Austria ; Schlein, Benjamin; Yau, Horng-Tzer
Abstract
We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are 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 first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales η ≫ N -1. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.
Publishing Year
Date Published
2010-01-01
Journal Title
International Mathematics Research Notices
Issue
3
Page
436 - 479
IST-REx-ID

### Cite this

Erdös L, Schlein B, Yau H. Wegner estimate and level repulsion for Wigner random matrices. International Mathematics Research Notices. 2010;(3):436-479. doi:10.1093/imrn/rnp136
Erdös, L., Schlein, B., & Yau, H. (2010). Wegner estimate and level repulsion for Wigner random matrices. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnp136
Erdös, László, Benjamin Schlein, and Horng Yau. “Wegner Estimate and Level Repulsion for Wigner Random Matrices.” International Mathematics Research Notices. Oxford University Press, 2010. https://doi.org/10.1093/imrn/rnp136.
L. Erdös, B. Schlein, and H. Yau, “Wegner estimate and level repulsion for Wigner random matrices,” International Mathematics Research Notices, no. 3. Oxford University Press, pp. 436–479, 2010.
Erdös L, Schlein B, Yau H. 2010. Wegner estimate and level repulsion for Wigner random matrices. International Mathematics Research Notices. (3), 436–479.
Erdös, László, et al. “Wegner Estimate and Level Repulsion for Wigner Random Matrices.” International Mathematics Research Notices, no. 3, Oxford University Press, 2010, pp. 436–79, doi:10.1093/imrn/rnp136.
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