Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra-Itzykson-Zuber Integral
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.
- Publication:
-
Annales Henri Poincaré
- Pub Date:
- May 2018
- DOI:
- 10.1007/s00023-018-0654-x
- arXiv:
- arXiv:1702.07100
- Bibcode:
- 2018AnHP...19.1307F
- Keywords:
-
- Mathematical Physics;
- Mathematics - Probability
- E-Print:
- 33 pages. To appear in Ann. Henri Poincare