Matrix Product Ensembles of Hermite Type and the Hyperbolic HarishChandraItzyksonZuber 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 biorthogonal ensemble, which reduces asymptotically to the Hermite MuttalibBorodin ensemble. Explicit expressions for the biorthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer Gfunctions, 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 socalled hyperbolic HarishChandraItzyksonZuber integral.
 Publication:

Annales Henri Poincaré
 Pub Date:
 May 2018
 DOI:
 10.1007/s000230180654x
 arXiv:
 arXiv:1702.07100
 Bibcode:
 2018AnHP...19.1307F
 Keywords:

 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 33 pages. To appear in Ann. Henri Poincare