Author(s):

Erdős, László; Schlein, Benjamin; Yau, HorngTzer

Article Title: 
Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

Affiliation 

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 ℓpnorms 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.

Keywords: 
Density of states; Wigner random matrix; Random Schrödinger operator; Semicircle law; localization; extended states

Journal Title:

Annals of Probability

Volume: 
37

Issue 
3

ISSN:

00911798

Publisher:

Institute of Mathematical Statistics

Date Published:

20090101

Start Page: 
815

End Page:

852

URL: 

DOI: 
10.1214/08AOP421

Open access: 
no 