Ewens Measures on Compact Groups and Hypergeometric Kernels

  • Paul Bourgade
  • Ashkan Nikeghbali
  • Alain RouaultEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

Decomposition of Haar measure Random matrices Characteristic polynomials Ewens sampling formula Correlation kernel 


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A.N.’s work is supported by the Swiss National Science Foundation (SNF) grant 200021_119970/1.

A.R’s work is partly supported by the ANR project Grandes Matrices Alatoires ANR-08-BLAN-0311-01.


  1. 1.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  2. 2.
    Andrews, G.E., Askey, R.A., Roy, R.: Special functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  3. 3.
    Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics, vol. 1. European Mathematical Society Publishing House, Zürich (2003)Google Scholar
  4. 4.
    Askey, R.A. (ed.): Gabor Szegö: Collected papers, vol. I. Birkhäuser, Basel (1982)Google Scholar
  5. 5.
    Basor, E.L., Chen, Y.: Toeplitz determinants from compatibility conditions. Ramanujan J. 16, 25–40 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Blower, G. Random matrices: high dimensional phenomena. London Mathematical Society Lecture Note Series, vol. 367. Cambridge University Press (2009)Google Scholar
  7. 7.
    Borodin, A., Olshanski, G.: Infinite random matrices and Ergodic measures. Commun. Math. Phys. 203, 87–123 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Borodin, A., Deift, P.: Fredholm determinants, Jimbo-Miwa-Ueno-functions, and representation theory. Commun. Pure Appl. Math. 55, 1160–1230 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Böttcher, A., Silbermann, B.: Toeplitz matrices and determinants with Fisher-Hartwig symbols. J. Funct. Anal. 63(2), 178–214 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bourgade, P., Hughes, C.P., Nikeghbali, A., Yor, M.: The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145(1), 45–69 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bourgade, P.: Conditional Haar measures on classical compact groups. Ann. Probab. 37(4), 1566–1586 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bourgade, P., Nikeghbali, A., Rouault, A.: Circular Jacobi ensembles and deformed Verblunski coefficients. Int. Math. Res. Not. 2009(23), 4357–4394 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bourgade, P.: A propos des matrices alatoires et des fonctions L. Thesis, ENST Paris (2009) available online at
  14. 14.
    Cohen, A.M.: Finite quaternionic reflection groups. J. Algebra 64(2), 293–324 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Diaconis, P., Shahshahani, M.: The subgroup algorithm for generating uniform random variables. Probab. Eng. Inform. Sci. 1, 15–32 (1987)zbMATHCrossRefGoogle Scholar
  16. 16.
    Forrester, P.J.: Log-Gases and Random Matrices, Book available online at
  17. 17.
    Hambly, B.M., Keevash, P., O’Connell, N., Stark, D.: The characteristic polynomial of a random permutation matrix. Stoch. Process. Appl. 90, 335–346 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Science Press, Peking (1958) Transl. Math. Monographs 6, Am. Math. Soc., 1963.Google Scholar
  19. 19.
    Katz, N.M., Sarnak, P.: Random Matrices, Frobenius Eigenvalues and Monodromy American Mathematical Society, vol. 45. Colloquium Publications (1999)Google Scholar
  20. 20.
    Katz, N.M., Sarnak, P.: Zeros of zeta functions and symmetry. Bull. Am. Soc. 36, 1–26 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Keating, J.P., Snaith, N.C.: Random matrix theory and \(\zeta (1/2 + it)\). Commun. Math. Phys. 214, 57–89 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Levin, E., Lubinsky, D.: Universality limits involving orthogonal polynomials on the unit circle, Comput. Meth. Funct. Theory 7, 543–561 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lubinsky, D.: Mutually Regular Measures have Similar Universality Limits. In: Neamtu, M., Schumaker, L.(eds.) Proceedings of 12th Texas Conference on Approximation Theory, pp. 256–269. Nashboro Press, Nashville (2008)Google Scholar
  24. 24.
    Mezzadri, F.: How to generate random matrices from the classical compact groups. Notices Am. Math. Soc. 54(5), 592–604 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Najnudel, J., Nikeghbali, A., Rubin, F.: Scaled limit and rate of convergence for the largest Eigenvalue from the generalized Cauchy random matrix ensemble. J. Stat. Phys. 137 (2009)Google Scholar
  26. 26.
    Neretin, Yu.A.: Hua type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114, 239–266 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Pickrell, D.: Measures on infinite-dimensional Grassmann manifolds. J. Funct. Anal. 70(2), 323–356 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Pickrell, D.: Mackey analysis of infinite classical motion groups. Pacific J. Math. 150, 139–166 (1991)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Pitman, J.: Combinatorial stochastic processes. Ecole d’Et de Probabilits (Saint-Flour, 2002), Lecture Notes in Mathematics, vol. 1875. Springer, (2006)Google Scholar
  30. 30.
    Rambour, Ph., Seghier, A.: Comportement asymptotique des polynmes orthogonaux associes un poids ayant un zro d’ordre fractionnaire sur le cercle. Applications aux valeurs propres d’une classe de matrices alatoires unitaires, (2009)
  31. 31.
    Simon, B.: The Christoffel–Darboux kernel, In: “Perspectives in PDE, Harmonic Analysis and Applications,” a volume in honor of V.G. Maz’ya’s 70th birthday, Proceedings of Symposia in Pure Mathematics, vol. 79, pp. 295–335 (2008)Google Scholar
  32. 32.
    Tsilevich, N.V.: Distribution of cycle lengths of infinite permutations. Zap. Nauchn. Sem. (POMI), 223, 148–161, 339 (1995). Translation in J. Math. Sci. 87(6), 4072–4081 (1997)Google Scholar
  33. 33.
    Wieand, K.: Permutation matrices, wreath products, and the distribution of eigenvalues. J. Theor. Probab. 16, 599–623 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Witte, N.S., Forrester, P.J.: Gap probabilities in the finite and scaled Cauchy random matrix ensembles. Nonlinearity, 13, 1965–1986 (2000)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Paul Bourgade
    • 1
  • Ashkan Nikeghbali
    • 2
  • Alain Rouault
    • 3
    Email author
  1. 1.Institut Telecom & Université Paris 6Paris Cedex 13France
  2. 2.Institut für MathematikUniversität ZürichZürichSwitzerland
  3. 3.Université Versailles-Saint Quentin, LMV, Bâtiment FermatVersailles CedexFrance

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