Determinantal structure of the overlaps for induced Ginibre / spherical unitary ensembles
A talk in the Seminar Zufallsmatrizen series by
Kohei Noda
| Abstract: | Recently, overlap, which is defined by the left and right eigenvectors of a
matrix, is one of the hottest topics in random matrix theory. This plays a role in
measuring the non-Hermiticity of the matrix. Indeed, the overlap is trivial for
Hermitian matrices, and hence, it plays an essential role for non-Hermitian
matrices. In 2020, Akemann, Tribe, Tsareas, and Zaboronski showed that the k-
th conditional expectation of the overlaps for the Ginibre unitary ensemble
forms a determinantal structure. In this talk, based on their approach, I will
show the determinantal structure of the overlaps for the induced
Ginibre/spherical unitary ensembles. The former model is the generalization
of the Ginibre unitary ensemble with the origin point insertion, and the latter
model is the non-Gaussian model with the origin point insertion. I will also
present the scaling limits for both models. The scaling limits in the strong non-
unitary regime are the same as those shown by Akemann et al for the Ginibre
unitary ensemble. As a consequence, the universality of the overlaps is
confirmed. On the other hand, I will present new scaling limits for the weakly
non-unitary regime and the singular origin regime. Please contact Lucas Hackl (Lucas.Hackl@unimelb.edu.au) for details regarding access Within the CRC this talk is associated to the project(s): C6 |