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Wegner Estimate and Level Repulsion for Wigner Random Matrices
Open accessAbstract
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.
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Cited by
Rigidity of eigenvalues of generalized Wigner matrices
Erdős, L., Yau, H.-T., & Yin, J. (2012) · Advances in Mathematics · doi:10.1016/j.aim.2011.12.010 · Cited by 345
Self-citationBulk universality for generalized Wigner matrices
Erdős, L., Yau, H.-T., & Yin, J. (2012) · Probability Theory and Related Fields · doi:10.1007/s00440-011-0390-3 · Cited by 254
Self-citationLocal law and Tracy–Widom limit for sparse random matrices
Lee, J. O., & Schnelli, K. (2018) · Probability Theory and Related Fields · doi:10.1007/s00440-017-0787-8 · Cited by 87
Metadata
- Authors
- Erdős, L., Schlein, B., & Yau, H.-T.
- Venue
- International Mathematics Research Notices
- Publisher
- Oxford University Press (OUP)
- Published
- 2009-08-29
- Volume
- 2010
- Issue
- 3
- Pages
- 436-479
- ISSN
- 1073-7928 (ISSN-L)
- 1687-0247
- OpenAlex
- W2963432094
- Cited by
- 169
- References
- 0
- FWCI
- 14.48
- Type
- journal article
- Language
- en
- Open access
- green
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- Erdős, L., Schlein, B., & Yau, H.-T. (2010). Wegner Estimate and Level Repulsion for Wigner Random Matrices. International Mathematics Research Notices. https://doi.org/10.1093/imrn/rnp136
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