Approaching Products Involving Hermitian Matrices with Harmonic Analysis
A talk in the AG Zufallsmatrizen series by
Mario Kieburg
| Abstract: | Please contact akemann@physik.uni-bielefeld.de for Meeting-ID and Password if you want to participate.
Products of random matrices have been successfully analytically studied from the viewpoint of harmonic analysis for the past five years. The reason of its feasibility has been born out of the explicit knowledge of some group integrals of the form of the Harish-Chandra integral resulting in determinantal point processes. Originally those products involved complex matrices and positive definite Hermitian matrices. Extending those to products involving arbitrary Hermitian matrices opened new possibilities in creating new local as well as global spectral statistics, but they have exhibited also analytical obstacles as shown by Forrester, Ipsen and Liu in their work considering products of complex induced Ginibre matrices with Hermitian matrices. Those obstacles are related to non-compact group structures due to the fact of positive and negative eigenvalues. Harmonic analysis helps to overcome these problems so that one can extend the result to non-Gaussian random matrices. I will report on this approach and which consequences the results yield. |