Skip to main content
SpringerLink
Log in

No-gaps delocalization for general random matrices

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its \({\ell_2}\) norm. Our results pertain to a wide class of random matrices, including matrices with independent entries, symmetric and skew-symmetric matrices, as well as some other naturally arising ensembles. The matrices can be real and complex; in the latter case we assume that the real and imaginary parts of the entries are independent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices. Cambridge Studies in Advanced Mathematics, Vol. 118. Cambridge University Press, Cambridge (2010).

  2. S. Arora and A. Bhaskara. Eigenvectors of random graphs: delocalization and nodal domains. (2011, manuscript).

  3. Z. Bai and J. Silverstein. Spectral analysis of large dimensional random matrices. 2nd edn. Springer Series in Statistics. Springer, New York (2010).

  4. Bai Z.D., Silverstein J., Yin. Y.: A note on the largest eigenvalue of a large dimensional sample covariance matrix. Journal of Multivariate Analysis 26, 166–168 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benaych-Georges F., Péché S.: Localization and delocalization for heavy tailed band matrices. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 50, 1385–1403 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bordenave C., Guionnet A.: Localization and delocalization of eigenvectors for heavy-tailed random matrices. Probability Theory and Related Fields 157, 885–953 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cacciapuoti C., Maltsev A., Schlein B.: Local Marchenko-Pastur law at the hard edge of sample covariance matrices. Journal of Mathematical Physics 54(4), 043302 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. K.R. Davidson and S. Szarek. Local operator theory, random matrices and Banach spaces. In: W.B. Johnson and J. Lindenstrauss (eds.) Handbook on the Geometry of Banach spaces, Vol. 1. Elsevier Science (2001), pp. 317–366.

  9. Dekel Y., Lee J.R., Linial N.: Eigenvectors of random graphs: nodal domains. Random Structures Algorithms 39, 39–58 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Eldan, M. Rácz and T. Schramm. Braess’s paradox for the spectral gap in random graphs and delocalization of eigenvectors. (2016, submitted). arXiv:1504.07669.

  11. L. Erdös. Universality for random matrices and log-gases. Current developments in mathematics, Vol. 2012. Int. Press, Somerville (2013), pp. 59–132.

  12. Erdös L., Knowles A.: Quantum diffusion and eigenfunction delocalization in a random band matrix model. Communications in Mathematical Physics 303, 509–554 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Erdös L., Knowles A.: Quantum diffusion and delocalization for band matrices with general distribution. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 12, 1227–1319 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Erdös L., Knowles A., Yau H.-T., Yin J.: Spectral statistics of Erdös-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics 314(3), 587–640 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Erdös, A. Knowles, H.-T. Yau and J. Yin. Spectral statistics of Erdös-Rényi graphs I: local semicircle law. Annals of Probability, (3B)41 (2013), 2279–2375.

  16. Erdös L., Knowles A., Yau H.-T., Yin J.: Delocalization and diffusion profile for random band matrices. Communications in Mathematical Physics 323, 367–416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Erdös L., Schlein B., Yau H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. The Annals of Probability 37, 815–852 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Erdös L., Schlein B., Yau H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics 287, 641–655 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Erdös L., Yau H.-T.: Universality of local spectral statistics of random matrices. Bulletin of the American Mathematical Society (N.S.) 49, 377–414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Gordon. On Dvoretzky’s theorem and extensions of Slepian’s lemma. Israel seminar on geometrical aspects of functional analysis (1983/84), Vol. II. Tel Aviv Univ., Tel Aviv, (1984), p. 25.

  21. Gordon Y.: Some inequalities for Gaussian processes and applications. The Israel Journal of Mathematics 50, 265–289 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kashin B.: The widths of certain finite-dimensional sets and classes of smooth functions. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 41, 334–351 (1977)

    MathSciNet  Google Scholar 

  23. Latala R.: Some estimates of norms of random matrices. Proceedings of the American Mathematical Society 133, 1273–1282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Litvak A., Pajor A., Rudelson M., Tomczak-Jaegermann N.: Smallest singular value of random matrices and geometry of random polytopes. Advances in Mathematics 195, 491–523 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Nguyen, T. Tao and V. Vu. Random matrices: tail bounds for gaps between eigenvalues. Probability Theory and Related Fields. doi:10.1007/s00440-016-0693-5.

  26. Rudelson M.: Invertibility of random matrices: norm of the inverse. Annals of Mathematics 168, 575–600 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Rudelson. Recent developments in non-asymptotic theory of random matrices, Modern aspects of random matrix theory. In: Proc. Sympos. Appl. Math., Vol. 72. Amer. Math. Soc., Providence, (2014), pp. 83–120.

  28. Rudelson M., Vershynin R.: The Littlewood-Offord Problem and invertibility of random matrices. Advances in Mathematics 218, 600–633 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Rudelson and R. Vershynin. The least singular value of a random square matrix is \({O(n^{-1/2})}\). Comptes rendus de l’Académie des sciences Mathématique, 346 (2008), 893–896.

  30. Rudelson M., Vershynin R.: Smallest singular value of a random rectangular matrix. Communications on Pure and Applied Mathematics 62, 1707–1739 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Rudelson and R. Vershynin. Non-asymptotic theory of random matrices: extreme singular values. In: Proceedings of the International Congress of Mathematicians, Vol. III. Hindustan Book Agency, New Delhi, (2010), 1576–1602.

  32. M. Rudelson and R. Vershynin. Small ball probabilities for linear images of high dimensional distributions. International Mathematics Research Notices, (19) 2015 (2015), 9594–9617. arXiv:1402.4492.

  33. M. Rudelson and R. Vershynin. Delocalization of eigenvectors of random matrices with independent entries. Duke Mathematical Journal, 164 (2015), 2507–2538. arXiv:1306.2887.

  34. E.M. Stein and G. Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, (1971).

  35. T. Tao. Topics in random matrix theory. Graduate Studies in Mathematics, Vol. 132. American Mathematical Society, Providence (2012).

  36. T. Tao and V. Vu. The condition number of a randomly perturbed matrix. In: STOC’07–Proceedings of the 39th Annual ACM Symposium on Theory of Computing. ACM, New York (2007), pp. 248–255.

  37. Tao T., Vu V.: Inverse Littlewood-Offord theorems and the condition number of random discrete matrices. Annals of Mathematics (2) 169, 595–632 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tao T., Vu V.: From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices. Bulletin of the American Mathematical Society (N.S.) 46, 377–396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tao T., Vu V.: Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur. The Annals of Probability 38, 2023–2065 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Tao and V. Vu, Random matrices: the universality phenomenon for Wigner ensembles. Modern aspects of random matrix theory. In: Proc. Sympos. Appl. Math., Vol. 72. Amer. Math. Soc., Providence (2014), pp. 121–172.

  41. Tran L.V., Vu V., Wang K.: Sparse random graphs: eigenvalues and eigenvectors. Random Structures Algorithms 42, 110–134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. R. Vershynin. Introduction to the non-asymptotic analysis of random matrices. Compressed sensing. Cambridge Univ. Press, Cambridge (2012), pp. 210–268. arXiv:1011.3027

  43. R. Vershynin. Invertibility of symmetric random matrices. Random Structures and Algorithms, 44 (2014), 135–182. arXiv:1102.0300.

  44. V. Vu and K. Wang. Random weighted projections, random quadratic forms and random eigenvectors. Random Structures and Algorithms. 47 (2015), 792–821. arXiv:1306.3099.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI, 48109, USA

    Mark Rudelson & Roman Vershynin

Authors
  1. Mark Rudelson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Vershynin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Rudelson.

Additional information

Partially supported by NSF grants DMS 1161372, 1265782, 1464514, and USAF Grant FA9550-14-1-0009.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rudelson, M., Vershynin, R. No-gaps delocalization for general random matrices. Geom. Funct. Anal. 26, 1716–1776 (2016). https://doi.org/10.1007/s00039-016-0389-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-016-0389-0

Search

Navigation