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Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

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Abstract

We consider Gaussian elliptic random matrices X of a size \(N \times N\) with parameter \(\rho \), i.e., matrices whose pairs of entries \((X_{ij}, X_{ji})\) are mutually independent Gaussian vectors with \(\mathbb {E}\,X_{ij} = 0\), \(\mathbb {E}\,X^2_{ij} = 1\) and \(\mathbb {E}\,X_{ij} X_{ji} = \rho \). We are interested in the asymptotic distribution of eigenvalues of the matrix \(W =\frac{1}{N^2} X^2 X^{*2}\). We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B:

$$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$

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References

  1. Akemann, G., Ipsen, J.R., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88(5), 052118 (2013)

    Article  Google Scholar 

  2. Alexeev, N., Götze, F., Tikhomirov, A.: Asymptotic distribution of singular values of powers of random matrices. Lith. Math. J. 50(2), 121–132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexeev, N., Götze, F., Tikhomirov, A.: On the asymptotic distribution of singular values of products of large rectangular random matrices. arXiv preprint arXiv:1012.2586 (2010)

  4. Alexeev, N., Götze, F., Tikhomirov, A.: On the singular spectrum of powers and products of random matrices. Dokl. Math. 82(1), 505–507 (2010)

    Article  MATH  Google Scholar 

  5. Arizmendi, O., Vargas, C.: Product of free random variables and k-divisible noncrossing partitions. Electron. Commun. Probab. 17, 1–13 (2012)

    Article  MATH  Google Scholar 

  6. Banica, T., Belinschi, S.T., Capitaine, M., Collins, B.: Free Bessel laws. Can. J. Math. 63(1), 3–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Biane, P., Goodman, F., Nica, A.: Non-crossing cumulants of type B. Trans. Am. Math. Soc. 355(6), 2263–2303 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burda, Z., Jarosz, A., Livan, G., Nowak, M.A., Swiech, A.: Eigenvalues and singular values of products of rectangular gaussian random matrices. Phys. Rev. E 82, 061114 (2010). doi:10.1103/PhysRevE.82.061114

  9. Fomin, S., Reading, N.: Root Systems and Generalized Associahedra. arXiv preprint arXiv:math/0505518 (2005)

  10. Forrester, P.J.: Eigenvalue statistics for product complex Wishart matrices. J. Phys. A Math. Theor. 47(34), 345202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Forrester, P.J., Liu, D.Z.: Raney distributions and random matrix theory. J. Stat. Phys. 158(5), 1051–1082 (2015). doi:10.1007/s10955-014-1150-4

  12. Girko, V.: The elliptic law. Teoriya Veroyatnostei i ee Primeneniya 30(4), 640–651 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Girko, V.: The strong elliptic law. Twenty years later. Part I. Random Oper. Stoch. Equ. 14(1), 59–102 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Götze, F., Naumov, A., Tikhomirov, A.: On minimal singular values of random matrices with correlated entries. Random Matrices: Theory and Applications 4(2), 1–30 (2015). doi:10.1142/S2010326315500069

  15. Götze, F., Naumov, A., Tikhomirov, A.: On one generalization of the elliptic law for random matrices. Acta Physica Polonica, Series B. 46(9), 1737–1745 (2015). doi:10.5506/APhysPolB.46.1737

  16. Lenczewski, R.: Limit distributions of random matrices. Adv. Math. 263, 253–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lenczewski, R., Salapata, R.: Multivariate Fuss-Narayana polynomials and their application to random matrices. Electron. J. Comb. 20(2), 1–14 (2013)

  18. Liu, D.Z., Song, C., Wang, Z.D.: On explicit probability densities associated with Fuss–Catalan numbers. Proc. Am. Math. Soc. 139(10), 3735–3738 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marchenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik 114(4), 507–536 (1967)

    Google Scholar 

  20. Mlotkowski, W.: Fuss-catalan numbers in noncommutative probability. Doc. Math. 15, 939–955 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Mlotkowski, W., Penson, K.A.: Probability distributions with binomial moments. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 17(02), 1450014 (2014). doi:10.1142/S0219025714500143

    Article  MathSciNet  MATH  Google Scholar 

  22. Mlotkowski, W., Penson, K.A., Zyczkowski, K.: Densities of the Raney distributions. Documenta Mathematica 18, 1573–1596 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Naumov, A.: Elliptic law for real random matrices. Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 31–38 (2013)

  24. Nguyen, H.H., O’Rourke, S.: The elliptic law. International Mathematics Research Notices p. rnu174 (2014)

  25. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability, vol. 13. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  26. O’Rourke, S., Renfrew, D.: Central limit theorem for linear eigenvalue statistics of elliptic random matrices. J. Theor. Probab. 1–71 (2014)

  27. O’Rourke, S., Renfrew, D., Soshnikov, A., Vu, V.: Products of independent elliptic random matrices. J. Stat. Phys. 160(1), 89–119 (2015). doi:10.1007/s10955-015-1246-5

  28. Penson, K.A., Zyczkowski, K.: Product of Ginibre matrices: Fuss–Catalan and Raney distributions. Phys. Rev. E 83(6), 061118 (2011)

    Article  Google Scholar 

  29. Reiner, V.: Non-crossing partitions for classical reflection groups. Discrete Math. 177(1), 195–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Speicher, R.: Polynomials in asymptotically free random matrices. arXiv preprint arXiv:1505.04337 (2015)

  31. The OEIS Foundation: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org (2015)

  32. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. 1. American Mathematical Society, Providence (1992)

    Book  MATH  Google Scholar 

  33. Zvonkin, A.: Matrix integrals and map enumeration: an accessible introduction. Math. Comput. Model. 26(8), 281–304 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zyczkowski, K., Penson, K.A., Nechita, I., Collins, B.: Generating random density matrices. J. Math. Phys. 52(6), 062201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are grateful for fruitful discussions with Pavel Galashin and Mikhail Basok. The authors thank the anonymous reviewer for valuable comments, which helped in improving the quality of the paper.

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Authors and Affiliations

  1. Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, Russia, 199178

    Nikita Alexeev

  2. George Washington University, Washington, D.C., USA

    Nikita Alexeev

  3. Department of Mathematics, Komi Science Center of Ural Division of RAS, Syktyvkar, Russia

    Alexander Tikhomirov

  4. Syktyvkar State University, Syktyvkar, Russia

    Alexander Tikhomirov

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  1. Nikita Alexeev
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  2. Alexander Tikhomirov
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Correspondence to Nikita Alexeev.

Additional information

N. Alexeev has been supported by the Grant of Russian Scientific Foundation 14-11-00581. A. Tikhomirov has been supported by SFB 701 Spectral Structures and Topological Methods in Mathematics University of Bielefeld, by Russian Foundation for Basic Research Grant 14-01-00500 and by Program of Fundamental Research Ural Division of RAS, Project N15-16-1-3.

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Alexeev, N., Tikhomirov, A. Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials. J Theor Probab 30, 1170–1190 (2017). https://doi.org/10.1007/s10959-016-0685-5

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  • DOI: https://doi.org/10.1007/s10959-016-0685-5

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