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

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Abstract

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.

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References

  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. Akemann, G., Ipsen, J.R.: Recent exact and asymptotic results for products of independent random matrices. Acta Phys. Pol. B 46, 1747 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Akemann, G., Ipsen, J.R., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013)

    Article  ADS  Google Scholar 

  4. Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart random matrices. J. Phys. A 46, 275205 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Akemann, G., Strahov, E.: Dropping the independence: singular values for products of two coupled random matrices. Commun. Math. Phys. 345, 101 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Alexeev, N., Götze, F., Tikhomirov, A.: Asymptotic distribution of singular values of powers of random matrices. Lith. Math. J. 50, 121 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andréief, C.: Note sur une relation les intégrales définies des produits des fonctions. Mém. de la Soc. Sci. Bordeaux 2 (1883)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Banica, T., Belinschi, S.T., Capitaine, M., Collins, B.: Free Bessel laws. Canad. J. Math. 63, 3 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Carlitz, L.: A note on certain biorthogonal polynomials. Pac. J. Math. 24, 425 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Claeys, T., Romano, S.: Biorthogonal ensembles with two-particle interactions. Nonlinearity 27, 2419 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Collar, A.R.: On the reciprocation of certain matrices. Proc. R. Soc. Edinb. 59, 195 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133 (1955)

    MathSciNet  MATH  Google Scholar 

  17. Desrosiers, P., Forrester, P.J.: Asymptotic correlations for Gaussian and Wishart matrices with external source. Int. Math. Res. Notices 2006, 27395 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Fields, J.L.: The asymptotic expansion of the Meijer G-function. Math. Comput. 26, 757–765 (1972)

    MathSciNet  MATH  Google Scholar 

  19. Forrester, P.J.: Log-gases and Random Matrices. Princeton University Press, Princeton, NJ (2010)

    MATH  Google Scholar 

  20. Forrester, P.J.: Eigenvalue statistics for product complex Wishart matrices. J. Phys. A 47, 345202 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Forrester, P.J., Ipsen, J.R.: Selberg integral theory and Muttalib–Borodin ensembles. arXiv:1612.06517

  22. Forrester, P.J., Liu, D.-Z.: Raney distributions and random matrix theory. J. Stat. Phys. 158, 1051 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Forrester, P.J., Liu, D.-Z., Zinn-Justin, P.: Equilibrium problems for Raney densities. Nonlinearity 28, 2265 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131, 1 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  29. Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87 (1957)

  30. Ipsen, J. R.: Products of independent Gaussian random matrices. PhD thesis, Bielefeld University (2015) (arXiv:1510.06128)

  31. Ipsen, J.R., Kieburg, M.: Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Phys. Rev. E 89, 032106 (2014)

    Article  ADS  Google Scholar 

  32. Itzykson, C., Zuber, J.-B.: The planar approximation. II. J. Math. Phys. 21, 411 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Kieburg, M., Kösters, H.: Exact relation between singular value and eigenvalue statistics. Random Matrices: Theor. Appl. 5, 1650015 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  36. Konhauser, J.D.E.: Biorthogonal polynomials suggested by the Laguerre polynomials. Pac. J. Math. 21, 303 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kuijlaars, A.B.J.: Transformations of polynomial ensembles. In: Modern Trends in Constructive Function Theory, vol. 253. American Mathematical Society (2016)

  38. Kuijlaars, A.B.J., Roman, P.: Spherical functions approach to sums of random Hermitian matrices. arXiv:1611.08932

  39. Kuijlaars, A.B.J., Stivigny, D.: Singular values of products of random matrices and polynomial ensembles. Random Matrices: Theor. Appl. 3, 1450011 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Kumar, S.: Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure. Phys. Rev. E 92, 032903 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  42. Liu, D.-Z.: Singular values for products of two coupled random matrices: hard edge phase transition. Constr Approx (2017). https://doi.org/10.1007/s00365-017-9389-z

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-Function: Theory and Applications. Springer, Berlin (2009)

    MATH  Google Scholar 

  45. Müller, R.R.: On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels. IEEE Trans. Inf. Theory 48, 2086 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Muttalib, K.A.: Random matrix models with additional interactions. J. Phys. A 28, L159 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  47. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  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)

  49. Pastur, L.A., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices. American Mathematical Society, Providence (2011)

    Book  MATH  Google Scholar 

  50. Penson, K.A., Zyczkowski, K.: Product of Ginibre matrices: Fuss–Catalan and Raney distributions. Phys. Rev. E 83, 061118 (2011)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  52. Simon, B.: The Christoffel–Darboux kernel. Proceedings of Symposia in Pure Mathematics 79, 295 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications, vol. 2. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  54. Wigner, E.: Statistical Properties of real symmetric matrices with many dimensions. In: Canadian Mathematical Congress Proceedings, University of Toronto Press 174 (1957)

  55. Zhang, L.: Local universality in biorthogonal Laguerre ensembles. J. Stat. Phys. 161, 688 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. ARC Centre of Excellence for Mathematical and Statistical Frontiers School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    P. J. Forrester & J. R. Ipsen

  2. Key Laboratory of Wu Wen-Tsun Mathematics CAS School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Dang-Zheng Liu

Authors
  1. P. J. Forrester
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  2. J. R. Ipsen
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  3. Dang-Zheng Liu
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Correspondence to J. R. Ipsen.

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Communicated by Vadim Gorin.

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Forrester, P.J., Ipsen, J.R. & Liu, DZ. Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral. Ann. Henri Poincaré 19, 1307–1348 (2018). https://doi.org/10.1007/s00023-018-0654-x

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