---
_id: '2770'
abstract:
- lang: eng
text: 'Consider N×N Hermitian or symmetric random matrices H with independent entries,
where the distribution of the (i,j) matrix element is given by the probability
measure vij with zero expectation and with variance σ ιj 2. We assume that the
variances satisfy the normalization condition Σiσij2=1 for all j and that there
is a positive constant c such that c≤Nσ ιj 2 ιc -1. We further assume that the
probability distributions νij have a uniform subexponential decay. We prove that
the Stieltjes transform of the empirical eigenvalue distribution of H is given
by the Wigner semicircle law uniformly up to the edges of the spectrum with an
error of order (Nη) -1 where η is the imaginary part of the spectral parameter
in the Stieltjes transform. There are three corollaries to this strong local semicircle
law: (1) Rigidity of eigenvalues: If γj=γj,N denotes the classical location of
the j-th eigenvalue under the semicircle law ordered in increasing order, then
the j-th eigenvalue λj is close to γj in the sense that for some positive constants
C, c P{double-struck}(∃j:|λ j-γ j|≥(logN) CloglogN[min(j,N-j+1)] -1/3N -2/3)≤
C exp[-(logN) cloglogN] for N large enough. (2) The proof of Dyson''s conjecture
(Dyson, 1962 [15]) which states that the time scale of the Dyson Brownian motion
to reach local equilibrium is of order N -1 up to logarithmic corrections. (3)
The edge universality holds in the sense that the probability distributions of
the largest (and the smallest) eigenvalues of two generalized Wigner ensembles
are the same in the large N limit provided that the second moments of the two
ensembles are identical.'
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Erdös L, Yau H, Yin J. Rigidity of eigenvalues of generalized Wigner matrices.
*Advances in Mathematics*. 2012;229(3):1435-1515. doi:10.1016/j.aim.2011.12.010
apa: Erdös, L., Yau, H., & Yin, J. (2012). Rigidity of eigenvalues of generalized
Wigner matrices. *Advances in Mathematics*. Academic Press. https://doi.org/10.1016/j.aim.2011.12.010
chicago: Erdös, László, Horng Yau, and Jun Yin. “Rigidity of Eigenvalues of Generalized
Wigner Matrices.” *Advances in Mathematics*. Academic Press, 2012. https://doi.org/10.1016/j.aim.2011.12.010.
ieee: L. Erdös, H. Yau, and J. Yin, “Rigidity of eigenvalues of generalized Wigner
matrices,” *Advances in Mathematics*, vol. 229, no. 3. Academic Press, pp.
1435–1515, 2012.
ista: Erdös L, Yau H, Yin J. 2012. Rigidity of eigenvalues of generalized Wigner
matrices. Advances in Mathematics. 229(3), 1435–1515.
mla: Erdös, László, et al. “Rigidity of Eigenvalues of Generalized Wigner Matrices.”
*Advances in Mathematics*, vol. 229, no. 3, Academic Press, 2012, pp. 1435–515,
doi:10.1016/j.aim.2011.12.010.
short: L. Erdös, H. Yau, J. Yin, Advances in Mathematics 229 (2012) 1435–1515.
date_created: 2018-12-11T11:59:30Z
date_published: 2012-02-15T00:00:00Z
date_updated: 2021-01-12T06:59:35Z
day: '15'
doi: 10.1016/j.aim.2011.12.010
extern: 1
intvolume: ' 229'
issue: '3'
month: '02'
page: 1435 - 1515
publication: Advances in Mathematics
publication_status: published
publisher: Academic Press
publist_id: '4120'
quality_controlled: 0
status: public
title: Rigidity of eigenvalues of generalized Wigner matrices
type: journal_article
volume: 229
year: '2012'
...