Our monthly meeting will take place on Monday the 31st of January, from 4pm to 6pm (Central European Time). Our speakers will be Leslie Molag and Gabriel Glesner. The zoom link will be sent to the members of the project through the mailing list. If you are interested in participating and you are not a PIICQ participant, please register sending us an email!
First talk by Leslie Molag at 4pm
Title: On the complex elliptic Ginibre ensemble and some generalizations.
Abstract: The complex elliptic Ginibre ensemble allows one to interpolate between the Ginibre ensemble and the Gaussian Unitary ensemble. It represents a determinantal point process in the complex plane with corresponding kernel, constructed with planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. It provides a unifying approach to rigorously derive several known and new results of local and global spectral statistics. In particular, we prove rigorously some global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. We introduce a d-complex dimensional generalization of the elliptic Ginibre ensemble, which interpolates between d-real and d-complex dimensions. In the Hermitian limit, this new ensemble is related to non-interacting Fermions in a trap in d-real dimensions with d-dimensional harmonic oscillator. We provide new local bulk and edge statistics at weak and strong non-Hermiticity for this new ensemble.
This is joint work with Gernot Akemann and Maurice Duits.
Second talk by Gabriel Glesner at 5pm
Title: Determinantal point processes conditioned on randomly incomplete configurations.
Abstract: We consider a marked point process with independent binomial marks 0 and 1. In our context of randomly incomplete configuration, we interpret the mark 1 points as detected while the mark 0 ones are unobserved. We then define the point process consisting of the undetected particles conditioned on a finite observation, i.e a finite configuration of mark 1 points.
When the ground process is determinantal, so is every one of the aforementioned point processes. Furthermore, important subclasses of determinantal point processes, namely the ones induced by projections and k-integrable kernels, are also preserved under this conditioning. In the latter case, the transformation can be characterised using a Riemann-Hilbert problem which can be seen as a combination of the celebrated method of Its, Izergin Korepin and Slavnov, with a discrete version of this method.
This is based on joint work with Tom Claeys (UCLouvain) https://arxiv.org/abs/2112.10642