## The Annals of Probability

- Ann. Probab.
- Volume 37, Number 3 (2009), 815-852.

### Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

László Erdős, Benjamin Schlein, and Horng-Tzer Yau

#### Abstract

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 *ℓ*^{p}-norms are comparable with *N*^{1/p−1/2} for *p*≥2, and we obtain the weaker bound *N*^{2/3(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.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 3 (2009), 815-852.

**Dates**

First available in Project Euclid: 19 June 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1245434021

**Digital Object Identifier**

doi:10.1214/08-AOP421

**Mathematical Reviews number (MathSciNet)**

MR2537522

**Zentralblatt MATH identifier**

1175.15028

**Subjects**

Primary: 15A52 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

**Keywords**

Semicircle law Wigner random matrix random Schrödinger operator density of states localization extended states

#### Citation

Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 (2009), no. 3, 815--852. doi:10.1214/08-AOP421. https://projecteuclid.org/euclid.aop/1245434021