Annales Henri Poincaré

, Volume 19, Issue 5, pp 1307–1348 | Cite as

Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral

  • P. J. Forrester
  • J. R. IpsenEmail author
  • Dang-Zheng Liu


We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib–Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra–Itzykson–Zuber integral.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akemann, G., Checinski, T., Liu, D.-Z., Strahov, E.: Finite rank perturbations in products of coupled random matrices: from one correlated to two Wishart ensembles. Ann. Inst. H. Poincaré Probab. Stat. (to appear) arXiv:1704.05224
  2. 2.
    Akemann, G., Ipsen, J.R.: Recent exact and asymptotic results for products of independent random matrices. Acta Phys. Pol. B 46, 1747 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Akemann, G., Ipsen, J.R., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart random matrices. J. Phys. A 46, 275205 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Akemann, G., Strahov, E.: Dropping the independence: singular values for products of two coupled random matrices. Commun. Math. Phys. 345, 101 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Alexeev, N., Götze, F., Tikhomirov, A.: Asymptotic distribution of singular values of powers of random matrices. Lith. Math. J. 50, 121 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andréief, C.: Note sur une relation les intégrales définies des produits des fonctions. Mém. de la Soc. Sci. Bordeaux 2 (1883)Google Scholar
  8. 8.
    Bai, Z.D., Miao, B., Jin, B.: On limit theorem for the eigenvalues of product of two random matrices. J. Multivar. Anal. 98, 76 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Banica, T., Belinschi, S.T., Capitaine, M., Collins, B.: Free Bessel laws. Canad. J. Math. 63, 3 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Burda, Z., Jarosz, A., Livan, G., Nowak, M.A., Swiech, A.: Eigenvalues and singular values of products of rectangular Gaussian random matrices. Phys. Rev. E 82, 061114 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Carlitz, L.: A note on certain biorthogonal polynomials. Pac. J. Math. 24, 425 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Claeys, T., Kuijlaars, A.B.J., Wang, D.: Correlation kernels for sums and products of random matrices. Random Matrices: Theory Appl. 4, 1550017 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Claeys, T., Romano, S.: Biorthogonal ensembles with two-particle interactions. Nonlinearity 27, 2419 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Collar, A.R.: On the reciprocation of certain matrices. Proc. R. Soc. Edinb. 59, 195 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133 (1955)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Desrosiers, P., Forrester, P.J.: Asymptotic correlations for Gaussian and Wishart matrices with external source. Int. Math. Res. Notices 2006, 27395 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fields, J.L.: The asymptotic expansion of the Meijer G-function. Math. Comput. 26, 757–765 (1972)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Forrester, P.J.: Log-gases and Random Matrices. Princeton University Press, Princeton, NJ (2010)zbMATHGoogle Scholar
  20. 20.
    Forrester, P.J.: Eigenvalue statistics for product complex Wishart matrices. J. Phys. A 47, 345202 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Forrester, P.J., Ipsen, J.R.: Selberg integral theory and Muttalib–Borodin ensembles. arXiv:1612.06517
  22. 22.
    Forrester, P.J., Liu, D.-Z.: Raney distributions and random matrix theory. J. Stat. Phys. 158, 1051 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Forrester, P.J., Liu, D.-Z.: Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition. Commun. Math. Phys. 344, 333 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Forrester, P.J., Liu, D.-Z., Zinn-Justin, P.: Equilibrium problems for Raney densities. Nonlinearity 28, 2265 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131, 1 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Forrester, P.J., Wang, D.: Muttalib–Borodin ensembles in random matrix theory—realisations and correlation functions. Electron. J. Prob. 22, paper no. 54 (2017)Google Scholar
  27. 27.
    Fyodorov, Y.V.: Negative moments of characteristic polynomials of random matrices: Ingham–Siegel integral as an alternative to Hubbard–Stratonovich transformation. Nucl. Phys. B 621, 643 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fyodorov, Y.V., Strahov, E.: Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds. Nucl. Phys. B 630, 453 (2002)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87 (1957)Google Scholar
  30. 30.
    Ipsen, J. R.: Products of independent Gaussian random matrices. PhD thesis, Bielefeld University (2015) (arXiv:1510.06128)
  31. 31.
    Ipsen, J.R., Kieburg, M.: Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Phys. Rev. E 89, 032106 (2014)ADSCrossRefGoogle Scholar
  32. 32.
    Itzykson, C., Zuber, J.-B.: The planar approximation. II. J. Math. Phys. 21, 411 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kieburg, M., Kösters, H.: Exact relation between singular value and eigenvalue statistics. Random Matrices: Theor. Appl. 5, 1650015 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kieburg, M., Kuijlaars, A.B.J., Stivigny, D.: Singular value statistics of matrix products with truncated unitary matrices. Int. Math. Res. Not. 2016, 3392 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kieburg, M., Verbaarschot, J.J.M., Zafeiropolous, S.: Spectral properties of the Wilson-Dirac operator and random matrix theory. Phys. Rev. D 88, 094502 (2013)ADSCrossRefGoogle Scholar
  36. 36.
    Konhauser, J.D.E.: Biorthogonal polynomials suggested by the Laguerre polynomials. Pac. J. Math. 21, 303 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kuijlaars, A.B.J.: Transformations of polynomial ensembles. In: Modern Trends in Constructive Function Theory, vol. 253. American Mathematical Society (2016)Google Scholar
  38. 38.
    Kuijlaars, A.B.J., Roman, P.: Spherical functions approach to sums of random Hermitian matrices. arXiv:1611.08932
  39. 39.
    Kuijlaars, A.B.J., Stivigny, D.: Singular values of products of random matrices and polynomial ensembles. Random Matrices: Theor. Appl. 3, 1450011 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kuijlaars, A.B.J., Zhang, L.: Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits. Commun. Math. Phys. 332, 759 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kumar, S.: Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure. Phys. Rev. E 92, 032903 (2015)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Liu, D.-Z.: Singular values for products of two coupled random matrices: hard edge phase transition. Constr Approx (2017).
  43. 43.
    Liu, D.-Z., Wang, D., Zhang, L.: Bulk and soft-edge universality for singular values of products of Ginibre random matrices. Ann. Inst. H. Poincaré Probab. Stat. 52, 1734 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-Function: Theory and Applications. Springer, Berlin (2009)zbMATHGoogle Scholar
  45. 45.
    Müller, R.R.: On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels. IEEE Trans. Inf. Theory 48, 2086 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Muttalib, K.A.: Random matrix models with additional interactions. J. Phys. A 28, L159 (1995)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  48. 48.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W., (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (Print companion to [DLMF]) (2010)Google Scholar
  49. 49.
    Pastur, L.A., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices. American Mathematical Society, Providence (2011)CrossRefzbMATHGoogle Scholar
  50. 50.
    Penson, K.A., Zyczkowski, K.: Product of Ginibre matrices: Fuss–Catalan and Raney distributions. Phys. Rev. E 83, 061118 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Pruisken, A.M.M., Schäfer, L.: The Anderson model for electron localisation non-linear sigma model, asymptotic gauge invariance. Nucl. Phys. B 200, 22 (1982)CrossRefGoogle Scholar
  52. 52.
    Simon, B.: The Christoffel–Darboux kernel. Proceedings of Symposia in Pure Mathematics 79, 295 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications, vol. 2. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  54. 54.
    Wigner, E.: Statistical Properties of real symmetric matrices with many dimensions. In: Canadian Mathematical Congress Proceedings, University of Toronto Press 174 (1957)Google Scholar
  55. 55.
    Zhang, L.: Local universality in biorthogonal Laguerre ensembles. J. Stat. Phys. 161, 688 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • P. J. Forrester
    • 1
  • J. R. Ipsen
    • 1
    Email author
  • Dang-Zheng Liu
    • 2
  1. 1.ARC Centre of Excellence for Mathematical and Statistical Frontiers School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Key Laboratory of Wu Wen-Tsun Mathematics CAS School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

Personalised recommendations