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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. Ipsen
  • Dang-Zheng Liu
Article

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • P. J. Forrester
    • 1
  • J. R. Ipsen
    • 1
  • 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

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