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

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. Ex...

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Bibliographic Details
Published inAnnales Henri Poincaré Vol. 19; no. 5; pp. 1307 - 1348
Main Authors Forrester, P. J., Ipsen, J. R., Liu, Dang-Zheng
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.05.2018
Springer Nature B.V
Subjects
Classical and Quantum Gravitation
Dynamical Systems and Ergodic Theory
Eigenvalues
Elementary Particles
Integrals
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Orthogonal functions
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
Transformations (mathematics)
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Summary: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.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-018-0654-x