Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Eigenvector distribution of Wigner matrices
Download PDF
Download PDF
  • Published: 14 December 2011

Eigenvector distribution of Wigner matrices

  • Antti Knowles1 &
  • Jun Yin1 

Probability Theory and Related Fields volume 155, pages 543–582 (2013)Cite this article

  • 642 Accesses

  • 64 Citations

  • 1 Altmetric

  • Metrics details

Abstract

We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Anderson G., Guionnet A., Zeitouni O.: An Introduction to Random Matrices Studies in advanced mathematics, 118. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. Bleher P., Its A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deift, P.: Orthogonal polynomials and random matrices: a Riemann–Hilbert approach. Courant Lecture Notes in Mathematics, vol. 3. American Mathematical Society, Providence (1999)

  4. Deift, P., Gioev, D.: Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics, vol. 18. American Mathematical Society, Providence (2009)

  5. Deift P., Kriecherbauer T., McLaughlin K.T.-R, Venakides S., Zhou X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deift P., Kriecherbauer T., McLaughlin K.T.-R, Venakides S., Zhou X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52, 1491–1552 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dyson F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  8. Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. arXiv:1103.3869 (2011, preprint)

  9. Erdős L., Schlein B., Yau H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37(3), 815–852 (2009)

    Article  MathSciNet  Google Scholar 

  10. Erdős L., Schlein B., Yau H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287, 641–655 (2009)

    Article  Google Scholar 

  11. Erdős L., Schlein B., Yau H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Notices 2010(3), 436–479 (2010)

    Google Scholar 

  12. Erdős, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. arXiv:0907.5605 (2011, to appear, preprint)

  13. Erdős, L., Ramirez, J., Schlein, B., Yau, H.-T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electr. J. Prob. 15, 526–604 (2010)

    Google Scholar 

  14. Erdős, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. H. Poincaré Probab. Stat. arXiv:0911.3687 (2011, to appear, preprint)

  15. Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. arXiv:1001.3453 (2011, preprint)

  16. Erdős, L., Yau, H.-T., Yin, J.: Universality for generalized Wigner matrices with Bernoulli distribution. J. Combin. arXiv:1003.3813 (2011, to appear, preprint)

  17. Erdős, L., Yau, H.-T., Yin, J.: Rigidity of eigenvalues of generalized Wigner matrices. arXiv:1007.4652 (2011, preprint)

  18. Gustavsson J.: Gaussian Fluctuations of Eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Stat. 41(2), 151–178 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Johansson K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions. arXiv:0910.4467 (2011, preprint)

  21. O’Rourke S.: Gaussian fluctuations of eigenvalues in Wigner random matrices. J. Stat. Phys. 138(6), 1045–1066 (2009)

    Article  MathSciNet  Google Scholar 

  22. Pastur L., Shcherbina M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130(2), 205–250 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sinai Y., Soshnikov A.: A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge. Funct. Anal. Appl. 32(2), 114–131 (1998)

    Article  MathSciNet  Google Scholar 

  24. Sodin, S.: The spectral edge of some random band matrices. arXiv:0906.4047 (2011, preprint)

  25. Soshnikov A.: Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207(3), 697–733 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tao, T., Vu, V.: Random matrices: universality of the local eigenvalue statistics. Acta Math. arXiv:0906.0510 (2011, to appear, preprint)

  27. Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. arXiv:0908.1982 (2011, preprint)

  28. Tao, T., Vu, V.: Random matrices: universal properties of eigenvectors. arXiv:1103.2801 (2011, preprint)

  29. Tracy C., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Antti Knowles & Jun Yin

Authors
  1. Antti Knowles
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antti Knowles.

Additional information

A. Knowles was partially supported by NSF grant DMS-0757425 and J. Yin was partially supported by NSF grant DMS-1001655.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Knowles, A., Yin, J. Eigenvector distribution of Wigner matrices. Probab. Theory Relat. Fields 155, 543–582 (2013). https://doi.org/10.1007/s00440-011-0407-y

Download citation

  • Received: 22 March 2011

  • Revised: 08 November 2011

  • Published: 14 December 2011

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-011-0407-y

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Random matrix
  • Universality
  • Eigenvector distribution

Mathematics Subject Classification (2010)

  • 15B52
  • 82B44
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature