  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 $U(\infty)$, 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.
