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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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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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Corresponding author

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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