Fast approach to the tracy-widom law at the edge of goe and gue1

Annals of Applied Probability

Volumn 22, Issue 5, 2012, Pages 1962-1988

  Johnstone, Iain M ^a   Ma, Zongming ^b  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

Largest eigenvalue; Random matrix; Rate of convergence

Indexed keywords

EID: 84879126948     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AAP819     Document Type: Article

References (25)

1. ADLER, M., FORRESTER, P. J., NAGAO, T. and VAN MOERBEKE, P. (2000). Classical skew orthogonal polynomials and random matrices. J. Stat. Phys. 99 141-170. MR1762659
2. ANDERSON, G. W., GUIONNET, A. and ZEITOUNI, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge. MR2760897
3. BORNEMANN, F. (2010). On the numerical evaluation of distributions in random matrix theory: A review. Markov Process. Related Field 16 803-866.
4. CHOUP, L. N. (2006). Edgeworth expansion of the largest eigenvalue distribution function of GUE and LUE. Int. Math. Res. Not. Art. ID 61049, 32. MR2233711
5. CHOUP, L. N. (2008). Edgeworth expansion of the largest eigenvalue distribution function of Gaussian unitary ensemble revisited. J. Math. Phys. 49 033508, 16. MR2406805
6. CHOUP, L. N. (2009). Edgeworth expansion of the largest eigenvalue distribution function of Gaussian orthogonal ensemble. J. Math. Phys. 50 013512, 22. MR2492622
7. EL KAROUI, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 2077-2117. MR2294977
8. FORRESTER, P. J. and MAYS, A. (2009). A method to calculate correlation functions for ? = 1 random matrices of odd size. J. Stat. Phys. 134 443-462. MR2485724
9. JOHNSTONE, I. M. (2008). Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy-Widom limits and rates of convergence. Ann. Statist. 36 2638-2716. MR2485010
10. JOHNSTONE, I. M. (2009). Approximate null distribution of the largest root in multivariate analysis. Ann. Appl. Stat. 3 1616-1633. MR2752150
11. MA, Z. (2012). Accuracy of the Tracy-Widom limits for the extreme eigenvalues in white Wishart matrices. Bernoulli 18 322-359.
12. OLVER, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York-London. MR0435697
13. PÉCHÉ, S. (2009). Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Related Fields 143 481-516. MR2475670
14. SEILER, E. and SIMON, B. (1975). An inequality among determinants. Proc. Natl. Acad. Sci. USA 72 3277-3278. MR0433247
15. SHI, W. (2008). A globally uniform asymptotic expansion of the Hermite polynomials. Acta Math. Sci. Ser. B Engl. Ed. 28 834-842. MR2462927
16. SINCLAIR, C. D. (2009). Correlation functions for ? = 1 ensembles of matrices of odd size. J. Stat. Phys. 136 17-33. MR2525224
17. SOSHNIKOV, A. (2002). A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108 1033-1056. MR1933444
18. SZEG?O, G. (1967). Orthogonal Polynomials, 3rd ed. Amer. Math. Soc., Providence, RI. MR0310533
19. TAO, T. and VU, V. (2010). Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 549-572. MR2669449
20. TRACY, C. A. and WIDOM, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. MR1257246
21. TRACY, C. A. and WIDOM, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727-754. MR1385083
22. TRACY, C. A. andWIDOM, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92 809-835. MR1657844
23. TRACY, C. A. andWIDOM, H. (2005). Matrix kernels for the Gaussian orthogonal and symplectic ensembles. Ann. Inst. Fourier (Grenoble) 55 2197-2207. MR2187952
24. WANG, X. S. and WONG, R. (2011). Asymptotics of orthogonal polynomials via recurrence relations. Technical report. Available at arXiv:1101.4371.
25. WONG, R. and ZHANG, L. (2007). Global asymptotics of Hermite polynomials via Riemann-Hilbert approach. Discrete Contin. Dyn. Syst. Ser. B 7 661-682 (electronic). MR2291866