---
_id: '2703'
abstract:
- lang: eng
text: '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. '
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Benjamin
full_name: Schlein, Benjamin
last_name: Schlein
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
citation:
ama: 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-AOP421
apa: Erdös, 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-AOP421
chicago: Erdös, 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.
ieee: 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.
ista: 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.
mla: 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.
short: L. Erdös, B. Schlein, H. Yau, Annals of Probability 37 (2009) 815–852.
date_created: 2018-12-11T11:59:09Z
date_published: 2009-01-01T00:00:00Z
date_updated: 2021-01-12T06:59:09Z
day: '01'
doi: 10.1214/08-AOP421
extern: 1
intvolume: ' 37'
issue: '3'
main_file_link:
- open_access: '0'
url: http://xxx.lanl.gov/abs/0711.1730
month: '01'
page: 815 - 852
publication: Annals of Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4193'
quality_controlled: 0
status: public
title: Semicircle law on short scales and delocalization of eigenvectors for Wigner
random matrices
type: journal_article
volume: 37
year: '2009'
...