# The Elliptic Tail Kernel

## TL;DR

This paper introduces a new family of $q$-translation-invariant determinantal point processes on the two-sided $q$-lattice, connecting $q$-deformed harmonic analysis with special functions and Young diagram structures.

## Contribution

It defines and analyzes a novel class of determinantal point processes, linking them to $q$-$zw$ measures and expressing their kernels via Jacobi theta functions.

## Key findings

- Processes are limits of $q$-$zw$ measures.
- Correlation kernels are expressed in terms of Jacobi theta functions.
- $q$-$zw$ measures are shown to be diffuse.

## 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 $U(\infty)$, 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.