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May 2009 Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
László Erdős, Benjamin Schlein, Horng-Tzer Yau
Ann. Probab. 37(3): 815-852 (May 2009). DOI: 10.1214/08-AOP421


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 ηN−2/3. We show that most eigenvectors are fully delocalized in the sense that their p-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.


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László Erdős. Benjamin Schlein. Horng-Tzer Yau. "Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices." Ann. Probab. 37 (3) 815 - 852, May 2009.


Published: May 2009
First available in Project Euclid: 19 June 2009

zbMATH: 1175.15028
MathSciNet: MR2537522
Digital Object Identifier: 10.1214/08-AOP421

Primary: 15A52 , 82B44

Keywords: density of states , extended states , Localization , random Schrödinger operator , semicircle law , Wigner random matrix

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 3 • May 2009
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