---
_id: '2758'
abstract:
- lang: eng
text: We consider N × N Hermitian random matrices with independent identical distributed
entries. The matrix is 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 prove that, away from the spectral edges, the density
of eigenvalues concentrates around the Wigner semicircle law on energy scales
n ≫ N -1 (log N) 8 . Up to the logarithmic factor, this is the smallest energy
scale for which the semicircle law may be valid. We also prove that for all eigenvalues
away from the spectral edges, the -tempℓ∞-norm of the corresponding eigenvectors
is of order O(N -1/2), modulo logarithmic corrections. The upper bound O(N -1/2)
implies that every eigenvector is completely delocalized, i.e., the maximum size
of the components of the eigenvector is of the same order as their average size.
In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions
on the distribution of the matrix elements.
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. Local semicircle law and complete delocalization
for Wigner random matrices. *Communications in Mathematical Physics*. 2009;287(2):641-655.
doi:10.1007/s00220-008-0636-9
apa: Erdös, L., Schlein, B., & Yau, H. (2009). Local semicircle law and complete
delocalization for Wigner random matrices. *Communications in Mathematical Physics*.
Springer. https://doi.org/10.1007/s00220-008-0636-9
chicago: Erdös, László, Benjamin Schlein, and Horng Yau. “Local Semicircle Law and
Complete Delocalization for Wigner Random Matrices.” *Communications in Mathematical
Physics*. Springer, 2009. https://doi.org/10.1007/s00220-008-0636-9.
ieee: L. Erdös, B. Schlein, and H. Yau, “Local semicircle law and complete delocalization
for Wigner random matrices,” *Communications in Mathematical Physics*, vol.
287, no. 2. Springer, pp. 641–655, 2009.
ista: Erdös L, Schlein B, Yau H. 2009. Local semicircle law and complete delocalization
for Wigner random matrices. Communications in Mathematical Physics. 287(2), 641–655.
mla: Erdös, László, et al. “Local Semicircle Law and Complete Delocalization for
Wigner Random Matrices.” *Communications in Mathematical Physics*, vol. 287,
no. 2, Springer, 2009, pp. 641–55, doi:10.1007/s00220-008-0636-9.
short: L. Erdös, B. Schlein, H. Yau, Communications in Mathematical Physics 287
(2009) 641–655.
date_created: 2018-12-11T11:59:27Z
date_published: 2009-04-01T00:00:00Z
date_updated: 2021-01-12T06:59:30Z
day: '01'
doi: 10.1007/s00220-008-0636-9
extern: 1
intvolume: ' 287'
issue: '2'
month: '04'
page: 641 - 655
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4134'
quality_controlled: 0
status: public
title: Local semicircle law and complete delocalization for Wigner random matrices
type: journal_article
volume: 287
year: '2009'
...