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
A concentration inequality and a local law for the sum of two random matrices
Download PDF
Download PDF
  • Published: 07 July 2011

A concentration inequality and a local law for the sum of two random matrices

  • Vladislav Kargin1 

Probability Theory and Related Fields volume 154, pages 677–702 (2012)Cite this article

  • 260 Accesses

  • 16 Citations

  • Metrics details

Abstract

Let \({H_{N}=A_{N}+U_{N}B_{N}U_{N}^{\ast}}\) where A N and B N are two N-by-N Hermitian matrices and U N is a Haar-distributed random unitary matrix, and let \({\mu _{H_{N}},}\) \({\mu_{A_{N}}, \mu _{B_{N}}}\) be empirical measures of eigenvalues of matrices H N , A N , and B N , respectively. Then, it is known (see Pastur and Vasilchuk in Commun Math Phys 214:249–286, 2000) that for large N, the measure \({\mu _{H_{N}}}\) is close to the free convolution of measures \({\mu _{A_{N}}}\) and \({\mu _{B_{N}}}\) , where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of \({\mu _{H_{N}}}\) from its expectation have been studied by Chatterjee (J Funct Anal 245:379–389, 2007). In this paper we improve Chatterjee’s concentration inequality and show that it holds with the rate which is quadratic in N. In addition, we prove a local law for eigenvalues of \({H_{N_{N}},}\) by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of μ A and μ B provided that the interval has width (log N)−1/2.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Anderson G.W., Guionnet A., Guionnet A., Guionnet A.: An introduction to random matrices In: Cambridge Studies in Advanced Mathematics, vol 118. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. Bai Z.D.: Convergence rate of expected spectral distributions of large random matrices, Part I Wigner matrices. Ann. Probab. 21, 625–648 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belinschi S.T.: The lebesgue decomposition of the free additive convolution of two probability distributions. Probab. Theory Relat. Fields 142, 125–150 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arous G.B., Guionnet A.: Large deviations for Wigner’s law and Voiculescu′s non-commutative entropy. Probab. Theory Relat. Fields 108, 517–542 (1997)

    Article  MATH  Google Scholar 

  5. Bercovici H., Voiculescu D.: Regularity questions for free convolutions. In: Bercovici, H., Foias, C (eds) Nonselfadjoint Operator Algebras, Operator Theory and Related Topics, Operator Theory Advances and Applications, vol. 104, pp. 37–47. Birkhauser, Basel (1998)

    Chapter  Google Scholar 

  6. Biane P.: Processes with free increments. Mathematische Zeitschrift 227, 143–174 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blower G.: Random Matrices: High Dimensional Phenomena volume 367 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  8. Chatterjee S.: Concentration of Haar measures with an application to random matrices. J. Funct. Anal. 245, 379–389 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Erdos, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. preprint arXiv:0911.3687 (2009)

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

    Article  MathSciNet  Google Scholar 

  11. Gromov M., Milman V.D.: A topological application of isoperimetric inequality. Am. J. Math. 105(4), 843–854 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hardy, G.H.: The mean value of the modulus of an analytic function. In: Proceedings of the London Mathematical Society, vol. 14, pp. 269–277 (1915)

  13. Horn A.: Eigenvalues of sums of Hermitian matrices. Pac. J. Math. 12, 225–241 (1962)

    MathSciNet  MATH  Google Scholar 

  14. Kantorovich, L.V.: Functional analysis and applied mathematics. Uspekhi Matematicheskih Nauk 3(6):89–185 (1948). English translation available in Kantorovich, L.V.: Selected Works, vol. 2, pp. 171–280. Gordon and Breach Science Publishers, New York (1996)

  15. Knutson A., Tao T.: The honeycomb model of \({GL_n(\mathbb{C})}\) tensor products I: proof of the saturation conjecture. J. Am. Math. Soc. 12, 1055–1090 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pastur L., Vasilchuk V.: On the law of addition of random matrices. Commun. Math. Phys. 214, 249–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Riesz, F.: Sur les valeurs moyennes du module des fonctions harmonique et des fonctions analytiques. Acta Litterarum ac Scientiarum, 1:27–32 (Available in vol. 1 of the collected papers by F. Riesz) (1922/23)

    Google Scholar 

  18. Speicher R.: Free convolution and the random sum of matrices. Publications of RIMS (Kyoto University) 29, 731–744 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. A.M.S. Providence (CRM Monograph series, No.1) (1992)

  20. Voiculescu D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Weyl H.: Das asymptotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1912)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Vladislav Kargin

Authors
  1. Vladislav Kargin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladislav Kargin.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kargin, V. A concentration inequality and a local law for the sum of two random matrices. Probab. Theory Relat. Fields 154, 677–702 (2012). https://doi.org/10.1007/s00440-011-0381-4

Download citation

  • Received: 22 November 2010

  • Revised: 09 June 2011

  • Published: 07 July 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0381-4

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

Mathematics Subject Classification (2010)

  • 60B20
  • 60B10
  • 46L54
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