$q$-deformed polyanalytic Ginibre ensemble
A talk in the Seminar Zufallsmatrizen series by
Yeong-Gwang Jung
| Abstract: | The Ginibre point process admits a description in terms of the lowest Landau level, and its extensions to higher Landau levels lead to polyanalytic Ginibre point processes. In this talk, I will introduce $q$-deformations of these models, constructed from a deformation of the canonical commutation relations. The resulting determinantal point processes are supported on concentric circles with geometrically spaced radii. I will then discuss their macroscopic distributions and fluctuations in the double-scaling regime $q=e^{-\lambda/N}$, where $\lambda>0$ is fixed and the system size $N$ tends to infinity. We obtain a $q$-deformed circular law with a non-uniform density on a disk and establish central limit theorems for smooth, compactly supported linear statistics. Although the deformation changes the macroscopic density, the limiting covariance, after rescaling the disk to unit size, coincides with that of the corresponding classical polyanalytic Ginibre ensemble. In particular, it is expressed through a Dirichlet seminorm in the bulk and an $H^{1/2}$ seminorm on the boundary. This talk is based on joint work with Ryosuke Sato. Within the CRC this talk is associated to the project(s): C6 |