Matrix product ensembles of Hermite-type

P. J. Forrester, J. R. Ipsen, Dang-Zheng Liu

Published 2017-02-23Version 1

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 find an explicit expression of the joint probability density function as a bi-orthogonal ensemble and it is shown that this ensemble reduces asymptotically to the Hermite Muttalib-Borodin model. Explicit expression for the bi-orthogonal functions as well as the correlation kernel are provided. Moreover, we find the explicit functional form of the local correlations near the origin (hard edge) and the global scaling limit of the spectrum. Both are compared with known results for the Hermite Muttalib-Borodin model. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself.