The Annals of Probability

The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble

Dong Wang

Full-text: Open access


We solve the largest sample eigenvalue distribution problem in the rank 1 spiked model of the quaternionic Wishart ensemble, which is the first case of a statistical generalization of the Laguerre symplectic ensemble (LSE) on the soft edge. We observe a phase change phenomenon similar to that in the complex case, and prove that the new distribution at the phase change point is the GOE Tracy–Widom distribution.

Article information

Ann. Probab., Volume 37, Number 4 (2009), 1273-1328.

First available in Project Euclid: 21 July 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]
Secondary: 60F99: None of the above, but in this section

Wishart distribution quaternionic spiked model


Wang, Dong. The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble. Ann. Probab. 37 (2009), no. 4, 1273--1328. doi:10.1214/08-AOP432.

Export citation


  • [1] Adler, M., Forrester, P. J., Nagao, T. and van Moerbeke, P. (2000). Classical skew orthogonal polynomials and random matrices. J. Statist. Phys. 99 141–170.
  • [2] Anderson, T. W. (1963). Asymptotic theory for principal component analysis. Ann. Math. Statist. 34 122–148.
  • [3] Andersson, S. (1975). Invariant normal models. Ann. Statist. 3 132–154.
  • [4] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • [5] Baik, J., Borodin, A., Rains, E. M. and Suidan, T. M. (2007). Unpublished results. In Conference Random and Integrable Models in Mathematics and Physics, September 11–15, Brussels.
  • [6] Baik, J. and Rains, E. M. (2001). The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 205–281.
  • [7] Baik, J. and Rains, E. M. (2001). Symmetrized random permutations. In Random Matrix Models and Their Applications. Mathematical Sciences Research Institute Publications 40 1–19. Cambridge Univ. Press, Cambridge.
  • [8] Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382–1408.
  • [9] de Bruijn, N. G. (1955). On some multiple integrals involving determinants. J. Indian Math. Soc. (N.S.) 19 133–151 (1956).
  • [10] Dumitriu, I., Edelman, A. and Shuman, G. (2007). MOPS: Multivariate orthogonal polynomials (symbolically). J. Symbolic Comput. 42 587–620.
  • [11] Forrester, P. J. (1993). The spectrum edge of random matrix ensembles. Nuclear Phys. B 402 709–728.
  • [12] Forrester, P. J. (2000). Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles. Available at
  • [13] Forrester, P. J. (2006). Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry. Nonlinearity 19 2989–3002.
  • [14] Forrester, P. J., Nagao, T. and Honner, G. (1999). Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Phys. B 553 601–643.
  • [15] Geman, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 252–261.
  • [16] Gohberg, I. C. and Kreĭn, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs 18. Amer. Math. Soc., Providence, RI. Translated from the Russian by A. Feinstein.
  • [17] Hanlon, P. J., Stanley, R. P. and Stembridge, J. R. (1992). Some combinatorial aspects of the spectra of normally distributed random matrices. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991). Contemporary Mathematics 138 151–174. Amer. Math. Soc., Providence, RI.
  • [18] Johansson, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 259–296.
  • [19] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • [20] Lax, P. D. (2002). Functional Analysis. Pure and Applied Mathematics (New York). Wiley, New York.
  • [21] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. Oxford Univ. Press, New York.
  • [22] Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 507–536.
  • [23] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Academic Press, Amsterdam.
  • [24] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • [25] Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 1617–1642.
  • [26] Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability 51 185–204. Birkhäuser, Boston.
  • [27] Stanley, R. P. (1989). Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 76–115.
  • [28] Szegő, G. (1975). Orthogonal Polynomials, 4th ed. Colloquium Publications XXIII. Amer. Math. Soc., Providence, RI.
  • [29] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • [30] Tracy, C. A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 727–754.
  • [31] Tracy, C. A. and Widom, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Statist. Phys. 92 809–835.