The Annals of Applied Probability

Fast approach to the Tracy–Widom law at the edge of GOE and GUE

Iain M. Johnstone and Zongming Ma

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We study the rate of convergence for the largest eigenvalue distributions in the Gaussian unitary and orthogonal ensembles to their Tracy–Widom limits.

We show that one can achieve an $O(N^{-2/3})$ rate with particular choices of the centering and scaling constants. The arguments here also shed light on more complicated cases of Laguerre and Jacobi ensembles, in both unitary and orthogonal versions.

Numerical work shows that the suggested constants yield reasonable approximations, even for surprisingly small values of $N$.

Article information

Ann. Appl. Probab., Volume 22, Number 5 (2012), 1962-1988.

First available in Project Euclid: 12 October 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 15B52: Random matrices

Rate of convergence random matrix largest eigenvalue


Johnstone, Iain M.; Ma, Zongming. Fast approach to the Tracy–Widom law at the edge of GOE and GUE. Ann. Appl. Probab. 22 (2012), no. 5, 1962--1988. doi:10.1214/11-AAP819.

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  • 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.
  • Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • Bornemann, F. (2010). On the numerical evaluation of distributions in random matrix theory: A review. Markov Process. Related Field 16 803–866.
  • Choup, L. N. (2006). Edgeworth expansion of the largest eigenvalue distribution function of GUE and LUE. Int. Math. Res. Not. Art. ID 61049, 32.
  • Choup, L. N. (2008). Edgeworth expansion of the largest eigenvalue distribution function of Gaussian unitary ensemble revisited. J. Math. Phys. 49 033508, 16.
  • Choup, L. N. (2009). Edgeworth expansion of the largest eigenvalue distribution function of Gaussian orthogonal ensemble. J. Math. Phys. 50 013512, 22.
  • El Karoui, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 2077–2117.
  • Forrester, P. J. and Mays, A. (2009). A method to calculate correlation functions for $\beta=1$ random matrices of odd size. J. Stat. Phys. 134 443–462.
  • Johnstone, I. M. (2008). Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy-Widom limits and rates of convergence. Ann. Statist. 36 2638–2716.
  • Johnstone, I. M. (2009). Approximate null distribution of the largest root in multivariate analysis. Ann. Appl. Stat. 3 1616–1633.
  • Ma, Z. (2012). Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices. Bernoulli 18 322–359.
  • Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York–London.
  • Péché, S. (2009). Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Related Fields 143 481–516.
  • Seiler, E. and Simon, B. (1975). An inequality among determinants. Proc. Natl. Acad. Sci. USA 72 3277–3278.
  • Shi, W. (2008). A globally uniform asymptotic expansion of the Hermite polynomials. Acta Math. Sci. Ser. B Engl. Ed. 28 834–842.
  • Sinclair, C. D. (2009). Correlation functions for $\beta=1$ ensembles of matrices of odd size. J. Stat. Phys. 136 17–33.
  • 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.
  • Szegő, G. (1967). Orthogonal Polynomials, 3rd ed. Amer. Math. Soc., Providence, RI.
  • Tao, T. and Vu, V. (2010). Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 549–572.
  • Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
  • Tracy, C. A. and Widom, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92 809–835.
  • Tracy, C. A. and Widom, H. (2005). Matrix kernels for the Gaussian orthogonal and symplectic ensembles. Ann. Inst. Fourier (Grenoble) 55 2197–2207.
  • Wang, X. S. and Wong, R. (2011). Asymptotics of orthogonal polynomials via recurrence relations. Technical report. Available at arXiv:1101.4371.
  • 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).