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(β), 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.Comment: 28 page