Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices

Duke Mathematical Journal

Volumn 133, Issue 2, 2006, Pages 205-235

  Baik, Jinho ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745619769     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-06-13321-5     Document Type: Article

References (30)

1. J. BAIK, G. BEN AROUS, and S. PÉCHÉ, Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, Ann. Probab. 33 (2005), 1643-1697. MR 2165575
2. J. BAIK, P. DEIFT, and E. STRAHOV, Products and ratios of characteristic polynomials of random Hermitian matrices, J. Math. Phys. 44 (2003), 3657-3670. MR 2006773
3. J. BAIK and E. M. RAINS, Limiting distributions for a polynuclear growth model with external sources, J. Statist. Phys. 100 (2000), 523-541. MR 1788477
4. _, The asymptotics of monotone subsequences of involutions, Duke Math. J. 109 (2001), 205-281. MR 1845180
5. J. BAIK and J. W. SILVERSTEIN, Eigenvalues of large sample covariance matrices of spiked population models, preprint, arXiv:math.ST/0408165
6. E. L. BASOR and H. WIDOM, On a Toeplitz determinant identity of Borodin and Okounkov, Integral Equations Operator Theory 37 (2000), 397-401. MR 1780119
7. A. BORODIN and A. OKOUNKOV, A Fredholm determinant formula for Toeplitz determinants. Integral Equations Operator Theory 37 (2000), 386-396. MR 1780118
8. A. BORODIN, A. OKOUNKOV, and G. OLSHANSKI, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481-515. MR 1758751
9. A. BÖTTCHER, "On the determinant formulas by Borodin, Okounkov, Baik, Deift and Rains" in Toeplitz Matrices and Singular Integral Equations (Pobershau, Germany, 2001), Oper. Theory Adv. Appl. 135, Birkhäuser, Basel, 2002. MR 1935759
10. P. DEIFT and X. ZHOU, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), 277-337. MR 1322812
11. A. FOKAS and X. ZHOU, On the solvability of Painlevé II and IV, Comm. Math. Phys. 144 (1992), 601-622. MR 1158763
12. P. J. FORRESTER, The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 (1993), 709-728. MR 1236195
13. _, Log-gases and random matrices, preprint, 2005, http://www.ms.unimelb.edu.au/~matpjf/matpjf.html
14. J. S. GERONIMO and K. M. CASE, Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys. 20 (1979), 299-310. MR 0519213
15. P. GLYNN and W. WHITT, Departures from many queues in series, Ann. Appl. Probab. 1 (1991), 546-572. MR 1129774
16. J. GRAVNER, C. TRACY, and H. WIDOM, Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Statist. Phys. 102 (2001), 1085-1132. MR 1830441
17. S. P. HASTINGS and J. B. MCLEOD, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), 31-51. MR 0555581
18. A. R. ITS and V. YU. NOVOKSHENOV, The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Lecture Notes in Math. 1191, Springer, Berlin, 1986. MR 0851569
19. K. JOHANSSON, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476. MR 1737991
20. _, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259-296. MR 1826414
21. I. M. JOHNSTONE, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist, 29 (2001), 295-327. MR 1863961
22. A. MARTÍNEZ-FINKELSHTEIN, K. T.-R. MCLAUGHLIN, and E. SAFF, Szegö orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics, preprint, arXiv:math.CA/0502300
23. V. B. MATVEEV and M. A. SALLE, Darboux Transformations and Solitons, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991. MR 1146435
24. D. PAUL, Asymptotic behaviour of the largest eigenvalues of sample covariance matrix when true covariance is a finite perturbation of identity, preprint, 2004.
25. M. PRÄHOFER, The Hastings-McLeod solution to Painlevé II and other tables, 2003, http://www-m5.ma.tum.de/pers/praehofer/
26. T. SEPPÄLÄINEN, A scaling limit for queues in series, Ann. Appl. Probab. 7 (1997), 855-872. MR 1484787
27. G. SZEGÖ, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, 1975. MR 0372517
28. C. TRACY and H. WIDOM, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174. MR 1257246
29. _, "Distribution functions for largest eigenvalues and their applications" in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed., Beijing, 2002, 587-596. MR 1989209
30. H. WIDOM, On convergence of moments for random Young tableaux and a random growth model, Int. Math. Res. Not. 2002, no. 9, 455-464. MR 1884467