The Annals of Probability

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

László Erdős, Benjamin Schlein, and Horng-Tzer Yau

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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 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.

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


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References

  • [1] Bai, Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 625–648.
  • [2] Bai, Z. D., Miao, B. and Tsay, J. (2002). Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J. 1 65–90.
  • [3] Bourgain, J. Private communication.
  • [4] Boutet de Monvel, A. and Khorunzhy, A. (1999). Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices. Random Oper. Stochastic Equations 7 149–168.
  • [5] Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366–389.
  • [6] Deift, P. A. (1999). Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. New York Univ. Courant Institute of Mathematical Sciences, New York.
  • [7] den Boer, A. F., van der Hofstad, R. and Klok, M. J. (2007). Large deviations for eigenvalues of sample covariance matrices. Preprint.
  • [8] Guionnet, A. (2009). Large Random Matrices: Lectures on Macroscopic Asymptotics. École d’Eté de Probabilités de Saint-Flour XXXVI 2006. Lecture Notes in Mathematics. Springer.
  • [9] Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 119–136.
  • [10] Johansson, K. (2001). Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 683–705.
  • [11] Khorunzhy, A. (1997). On smoothed density of states for Wigner random matrices. Random Oper. Stochastic Equations 5 147–162.