Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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

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.

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

References

  1. 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)

  2. 2

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

  3. 3

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

  4. 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)

  5. 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)

  6. 6

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

  7. 7

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

  8. 8

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

  9. 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. 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)

  11. 11

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

  12. 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. 13

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

  14. 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. 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)

  16. 16

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

  17. 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)

  18. 18

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

  19. 19

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

  20. 20

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

  21. 21

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

Download references

Author information

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

Mathematics Subject Classification (2010)

  • 60B20
  • 60B10
  • 46L54