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

Holger Kösters

Open Mathematics (2010)

  • Volume: 8, Issue: 4, page 763-779
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Holger Kösters. "Characteristic polynomials of sample covariance matrices: The non-square case." Open Mathematics 8.4 (2010): 763-779. <http://eudml.org/doc/269406>.

@article{HolgerKösters2010,
abstract = {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.},
author = {Holger Kösters},
journal = {Open Mathematics},
keywords = {Random matrices; Characteristic polynomials; Bessel functions; random matrices; characteristic polynomials; complex sample covariance matrix; Hamiltonian matrix; real sample covariance matrix; sine kernel; spectrum; Airy kernel},
language = {eng},
number = {4},
pages = {763-779},
title = {Characteristic polynomials of sample covariance matrices: The non-square case},
url = {http://eudml.org/doc/269406},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Holger Kösters
TI - Characteristic polynomials of sample covariance matrices: The non-square case
JO - Open Mathematics
PY - 2010
VL - 8
IS - 4
SP - 763
EP - 779
AB - 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.
LA - eng
KW - Random matrices; Characteristic polynomials; Bessel functions; random matrices; characteristic polynomials; complex sample covariance matrix; Hamiltonian matrix; real sample covariance matrix; sine kernel; spectrum; Airy kernel
UR - http://eudml.org/doc/269406
ER -

References

top
  1. [1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1965 Zbl0171.38503
  2. [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)00458-9[Crossref] Zbl1024.82012
  3. [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/10.1063/1.1587875[Crossref] Zbl1062.15014
  4. [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.20070[Crossref] Zbl1075.62014
  5. [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.20092[Crossref] Zbl1155.15304
  6. [6] Brézin E., Hikami S., Characteristic polynomials of random matrices, Comm. Math. Phys., 2000, 214(1), 111–135 http://dx.doi.org/10.1007/s002200000256[Crossref] Zbl1042.82017
  7. [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/s002200100547[Crossref] Zbl0987.15012
  8. [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/10.1088/0305-4470/36/3/309[Crossref] Zbl1066.82022
  9. [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 Zbl0997.47033
  10. [10] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Tables of Integral Transforms, vol. I, McGraw-Hill, New York, 1954 Zbl0055.36401
  11. [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 [WoS][Crossref] Zbl1198.60011
  12. [12] Forrester P.J., Log-Gases and Random Matrices, book in preparation, www.ms.unimelb.edu.au/ matpjf/matpjf.html Zbl1217.82003
  13. [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 Zbl1044.81050
  14. [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-z[WoS][Crossref] Zbl1193.15035
  15. [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 [Crossref] Zbl1189.60019
  16. [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 Zbl1213.60023
  17. [17] Kösters H., Characteristic polynomials of sample covariance matrices, J. Theoret. Probab., (in press), preprint available at http://arxiv.org/abs/0906.2763 Zbl1250.62034
  18. [18] Mehta M.L., Random Matrices, 3rd ed., Pure and Applied Mathematics, 142, Elsevier, Amsterdam, 2004 
  19. [19] Olver F.W.J., Asymptotics and Special Functions, Academic Press, New York, 1974 Zbl0303.41035
  20. [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/s00440-007-0133-7[WoS][Crossref] 
  21. [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:1019739414239[Crossref] Zbl1018.62042
  22. [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 Zbl1098.82018
  23. [23] Szegö G., Orthogonal Polynomials, 3rd ed., American Mathematical Society Colloquium Publications, 23, American Mathematical Society, Providence, 1967 
  24. [24] Tao T., Vu V., Random covariance matrices: universality of local statistics of eigenvalues, preprint available at http://arxiv.org/abs/0912.0966 Zbl1247.15036
  25. [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/s00220-004-1234-0[Crossref] Zbl1070.82013

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.