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.

中文翻译：

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椭圆尾核

我们在两侧的 $q$-晶格上介绍并研究了一个新的 $q$-平移不变行列式点过程。我们证明这些过程是 $q$-$zw$ 测度的极限，它出现在 $U(\infty)$ 谐波分析的 $q$-变形中，并用 Jacobi theta 函数表示它们的相关核. 作为一个应用程序，我们表明 $q$-$zw$ 度量是扩散的。我们的结果还暗示了双面 $q$-lattice 和 Young 图的行/列之间的联系。