Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
arXiv:1702.07100 · doi:10.1007/s00023-018-0654-x
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.
33 pages. To appear in Ann. Henri Poincare
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