The Elliptic Tail Kernel. (3rd March 2020) - Cuenca, Cesar; Gorin, Vadim; Olshanski, Grigori
Abstract: We introduce and study a new family of $q$ -translation-invariant determinantal point processes on the two-sided $q$ -lattice. We prove that these processes are limits of the $q$ –$zw$ measures, which arise in the $q$ -deformation of harmonic analysis on $U(∞ )$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $q$ –$zw$ measures are diffuse. Our results also hint at a link between the two-sided $q$ -lattice and rows/columns of Young diagrams.