Journal article
Matrix Product Ensembles of Hermite Type and the Hyperbolic Harish-Chandra–Itzykson–Zuber Integral
PJ Forrester, JR Ipsen, DZ Liu
Annales Henri Poincare | SPRINGER INTERNATIONAL PUBLISHING AG | Published : 2018
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 ..
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Awarded by Seventh Framework Programme
Funding Acknowledgements
We thank S. Kumar for useful discussions on his paper and M. Kieburg for comments on a first draft. We acknowledge support by the Australian Research Council through Grant DP170102028 (PJF), the ARC Centre of Excellence for Mathematical and Statistical Frontiers (PJF, JRI), and partially by ERC Advanced Grant #338804, the National Natural Science Foundation of China #11771417, the Youth Innovation Promotion Association CAS #2017491, the Fundamental Research Funds for the Central Universities #WK0010450002, Anhui Provincial Natural Science Foundation #1708085QA03 (DZL). D.-Z. Liu is particularly grateful to Laszlo Erdos for funding his one-year stay at IST Austria.