Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
See all formats and pricing
More options …
Volume 8, Issue 4


Volume 13 (2015)

Characteristic polynomials of sample covariance matrices: The non-square case

Holger Kösters
Published Online: 2010-07-24 | DOI: https://doi.org/10.2478/s11533-010-0035-2


We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices.

MSC: 15A52

Keywords: Random matrices; Characteristic polynomials; Bessel functions

  • [1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1965 Google Scholar

  • [2] Akemann G., Fyodorov Y.V., Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges, Nuclear Phys. B, 2003, 664(3), 457–476 http://dx.doi.org/10.1016/S0550-3213(03)00458-9CrossrefGoogle Scholar

  • [3] Baik J., Deift P., Strahov E., Products and ratios of characteristic polynomials of random Hermitian matrices. Integrability, topological solitons and beyond, J. Math. Phys., 2003, 44(8), 3657–3670 http://dx.doi.org/10.1063/1.1587875CrossrefGoogle Scholar

  • [4] Ben Arous G., Péché S., Universality of local eigenvalue statistics for some sample covariance matrices, Comm. Pure Appl. Math., 2005, 58(10), 1316–1357 http://dx.doi.org/10.1002/cpa.20070CrossrefGoogle Scholar

  • [5] Borodin A., Strahov E., Averages of characteristic polynomials in random matrix theory, Comm. Pure Appl. Math., 2006, 59(2), 161–253 http://dx.doi.org/10.1002/cpa.20092CrossrefGoogle Scholar

  • [6] Brézin E., Hikami S., Characteristic polynomials of random matrices, Comm. Math. Phys., 2000, 214(1), 111–135 http://dx.doi.org/10.1007/s002200000256CrossrefGoogle Scholar

  • [7] Brézin E., Hikami S., Characteristic polynomials of real symmetric random matrices, Comm. Math. Phys., 2001, 223(2), 363–382 http://dx.doi.org/10.1007/s002200100547CrossrefGoogle Scholar

  • [8] Brézin E., Hikami S., New correlation functions for random matrices and integrals over supergroups, J. Phys. A, 2003, 36(3), 711–751 http://dx.doi.org/10.1088/0305-4470/36/3/309CrossrefGoogle Scholar

  • [9] Deift, P.A., Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, 3, Courant Institute of Mathematical Sciences, New York, 1999 Google Scholar

  • [10] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Tables of Integral Transforms, vol. I, McGraw-Hill, New York, 1954 Google Scholar

  • [11] Feldheim O.N., Sodin S., A universality result for the smallest eigenvalues of certain sample covariance matrices, Geom. Funct. Anal., (in press), DOI:10.1007/s00039-010-0055-x Web of ScienceCrossrefGoogle Scholar

  • [12] Forrester P.J., Log-Gases and Random Matrices, book in preparation, www.ms.unimelb.edu.au/ matpjf/matpjf.html Google Scholar

  • [13] Fyodorov Y.V., Strahov E., An exact formula for general spectral correlation function of random Hermitian matrices, J. Phys. A, 2003, 36(12), 3202–3213 Google Scholar

  • [14] Götze F., Kösters H., On the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix, Comm. Math. Phys., 2009, 285(3), 1183–1205 http://dx.doi.org/10.1007/s00220-008-0544-zWeb of ScienceCrossrefGoogle Scholar

  • [15] Kösters H., On the second-order correlation function of the characteristic polynomial of a real symmetric Wigner matrix, Electron. Commun. Prob., 2008, 13, 435–447 CrossrefGoogle Scholar

  • [16] Kösters H., Asymptotics of characteristic polynomials of Wigner matrices at the edge of the spectrum, Asymptot. Anal., (in press), preprint available at http://arxiv.org/abs/0805.3044 Google Scholar

  • [17] Kösters H., Characteristic polynomials of sample covariance matrices, J. Theoret. Probab., (in press), preprint available at http://arxiv.org/abs/0906.2763 Google Scholar

  • [18] Mehta M.L., Random Matrices, 3rd ed., Pure and Applied Mathematics, 142, Elsevier, Amsterdam, 2004 Google Scholar

  • [19] Olver F.W.J., Asymptotics and Special Functions, Academic Press, New York, 1974 Google Scholar

  • [20] Péché, S., Universality results for the largest eigenvalues of some sample covariance matrix ensembles, Probab. Theory Related Fields, 2009, 143(3–4), 481–516 http://dx.doi.org/10.1007/s00440-007-0133-7Web of ScienceCrossrefGoogle Scholar

  • [21] Soshnikov A., A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices, J. Statist. Phys., 2002, 108(5–6), 1033–1056 http://dx.doi.org/10.1023/A:1019739414239CrossrefGoogle Scholar

  • [22] Strahov E., Fyodorov Y.V., Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach, Comm. Math. Phys., 2003, 241(2–3), 343–382 Google Scholar

  • [23] Szegö G., Orthogonal Polynomials, 3rd ed., American Mathematical Society Colloquium Publications, 23, American Mathematical Society, Providence, 1967 Google Scholar

  • [24] Tao T., Vu V., Random covariance matrices: universality of local statistics of eigenvalues, preprint available at http://arxiv.org/abs/0912.0966 Google Scholar

  • [25] Vanlessen M., Universal behavior for averages of characteristic polynomials at the origin of the spectrum, Comm. Math. Phys., 2003, 253(3), 535–560 http://dx.doi.org/10.1007/s00220-004-1234-0CrossrefGoogle Scholar

About the article

Published Online: 2010-07-24

Published in Print: 2010-08-01

Citation Information: Open Mathematics, Volume 8, Issue 4, Pages 763–779, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-010-0035-2.

Export Citation

© 2010 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Tatyana Shcherbina
Communications in Mathematical Physics, 2014, Volume 328, Number 1, Page 45
T. Shcherbina
Probability Theory and Related Fields, 2013, Volume 156, Number 1-2, Page 449

Comments (0)

Please log in or register to comment.
Log in