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Characteristic polynomials of sample covariance matrices. (English) Zbl 1250.62034

The author investigates the second-order correlation function of the characteristic polynomial of a sample covariance matrix. The results are valid in the complex and real setting, in the bulk , the soft edge and the hard edge of the spectrum. He obtains an explicit expression for an exponential-type generating function of the second-order correlation function of the characteristic polynomial. He recovers the well-known kernels of random matrix theory by asymptotic analysis.

MSC:

62H99 Multivariate analysis
15B52 Random matrices (algebraic aspects)
60E10 Characteristic functions; other transforms
62H20 Measures of association (correlation, canonical correlation, etc.)
62E20 Asymptotic distribution theory in statistics
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[1] Akemann, G., Fyodorov, Y.V.: Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges. Nucl. Phys. B 664, 457–476 (2003) · Zbl 1024.82012 · doi:10.1016/S0550-3213(03)00458-9
[2] Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 2nd edn. Wiley, New York (1984) · Zbl 0651.62041
[3] Baik, J., Deift, P., Strahov, E.: Products and ratios of characteristic polynomials of random Hermitian matrices. J. Math. Phys. 44, 3657–3670 (2003) · Zbl 1062.15014 · doi:10.1063/1.1587875
[4] Ben Arous, G., Péché, S.: Universality of local eigenvalue statistics for some sample covariance matrices. Commun. Pure Appl. Math. 58, 1316–1357 (2005) · Zbl 1075.62014 · doi:10.1002/cpa.20070
[5] Borodin, A., Strahov, E.: Averages of characteristic polynomials in random matrix theory. Commun. Pure Appl. Math. 59, 161–253 (2006) · Zbl 1155.15304 · doi:10.1002/cpa.20092
[6] Brézin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111–135 (2000) · Zbl 1042.82017 · doi:10.1007/s002200000256
[7] Brézin, E., Hikami, S.: Characteristic polynomials of real symmetric random matrices. Commun. Math. Phys. 223, 363–382 (2001) · Zbl 0987.15012 · doi:10.1007/s002200100547
[8] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of Integral Transforms, vol. I. McGraw-Hill, New York (1954) · Zbl 0055.36401
[9] Forrester, P.J.: Log Gases and Random Matrices (2008, in preparation). www.ms.unimelb.edu.au/\(\sim\)matpjf/matpjf.html · Zbl 1217.82003
[10] Forrester, P.J., Gamburd, A.: Counting formulas associated with some random matrix averages. J. Comb. Theory, Ser. A 113, 934–951 (2006) · Zbl 1097.15019 · doi:10.1016/j.jcta.2005.09.001
[11] Fyodorov, Y.V., Strahov, E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A, Math. Gen. 36, 3202–3213 (2003) · Zbl 1044.81050
[12] Götze, F., Kösters, H.: On the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix. Commun. Math. Phys. 285, 1183–1205 (2009) · Zbl 1193.15035 · doi:10.1007/s00220-008-0544-z
[13] Keating, J.P., Snaith, N.C.: Random matrix theory and {\(\zeta\)}(1/2+it). Commun. Math. Phys. 214, 57–89 (2000) · Zbl 1051.11048 · doi:10.1007/s002200000261
[14] Keating, J.P., Snaith, N.C.: Random matrix theory and L-functions at s=1/2. Commun. Math. Phys. 214, 91–110 (2000) · Zbl 1051.11047 · doi:10.1007/s002200000262
[15] Kösters, H.: On the second-order correlation function of the characteristic polynomial of a real-symmetric Wigner matrix. Electron. Commun. Probab. 13, 435–447 (2008) · Zbl 1189.60019 · doi:10.1214/ECP.v13-1400
[16] Kösters, H.: Asymptotics of characteristic polynomials of Wigner matrices at the edge of the spectrum. Asymptot. Anal. (2009, to appear) · Zbl 1213.60023
[17] Mehta, M.L.: Random Matrices, 3rd edn. Pure and Applied Mathematics, vol. 142, Elsevier, Amsterdam (2004) · Zbl 1107.15019
[18] Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982) · Zbl 0556.62028
[19] Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974) · Zbl 0303.41035
[20] Péché, S.: Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Relat. Fields 143, 481–516 (2009) · Zbl 1167.62019 · doi:10.1007/s00440-007-0133-7
[21] Péché, S.: Universality in the bulk of the spectrum for complex sample covariance matrices. Preprint (2009). arXiv: 0912.2493
[22] Ruzmaikina, A.: Universality of the edge distribution of eigenvalues of Wigner matrices with polynomially decaying distributions of entries. Commun. Math. Phys. 261, 277–296 (2006) · Zbl 1130.82313 · doi:10.1007/s00220-005-1386-6
[23] Soshnikov, A.: A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108, 1033–1056 (2002) · Zbl 1018.62042 · doi:10.1023/A:1019739414239
[24] Strahov, E., Fyodorov, Y.V.: Universal results for correlations of characteristic polynomials: Riemann–Hilbert approach. Commun. Math. Phys. 241, 343–382 (2003) · Zbl 1098.82018
[25] Szegö, G.: Orthogonal Polynomials, 3rd edn. American Mathematical Society Colloquium Publications, vol. XXIII. American Mathematical Society, Providence (1967) · JFM 65.0278.03
[26] Tao, T., Vu, V.: Random matrices: The distribution of the smallest singular values. Preprint (2009). arXiv: 0903.0614 · Zbl 1210.60014
[27] Tao, T., Vu, V.: Random covariance matrices: Universality of local statistics of eigenvalues. Preprint (2009). arXiv: 0912.0966 · Zbl 1247.15036
[28] Vanlessen, M.: Universal behavior for averages of characteristic polynomials at the origin of the spectrum. Commun. Math. Phys. 253, 535–560 (2003) · Zbl 1070.82013 · doi:10.1007/s00220-004-1234-0
[29] Vanlessen, M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. Constr. Approx. 25, 125–175 (2007) · Zbl 1117.15025 · doi:10.1007/s00365-005-0611-z
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