We consider the generating function of the sine point process on $m$ consecutive intervals. It can be written as a Fredholm determinant with discontinuities, or equivalently as the convergent series

$$\sum _{k_1,\dots ,k_m⩾0}\mathbb{P}\left({\cap }_{j=1}^m\#\{\text{points}\text{in}\text{the}\text{jth}\text{interval}\}=k_j\right)\prod _{j=1}^m{s}_j^{k_j},$$

where $s_1,\dots ,s_m\in [0,+\infty )$. In particular, we can deduce from it joint probabilities of the counting function of the process. In this work, we obtain large gap asymptotics for the generating function, which are asymptotics as the size of the intervals grows. Our results are valid for an arbitrary integer $m$, in the cases where all the parameters $s_1,\dots ,s_m$, except possibly one, are positive. This generalizes two known results: (1) a result of Basor and Widom, which corresponds to $m=1$ and $s_1>0$, and (2) the case $m=1$ and $s_1=0$ for which many authors have contributed. We also present some applications in the context of thinning and conditioning of the sine process.

中文翻译：

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正弦点过程生成函数的大间隙渐近

我们考虑正弦点过程的生成函数 $米$连续间隔。它可以写成一个具有不连续性的 Fredholm 行列式，或者等价地写成收敛级数

$$\sum _{{克}_1,\dots ,{克}_{米}⩾0}\mathbb{磷}\left({\cap }_{j=1}^{米}\#\{\text{积分}\text{在}\text{这}\text{第一个}\text{间隔}\}={克}_j\right)\prod _{j=1}^{米}{秒}_j^{{克}_j},$$

在哪里 ${秒}_1,\dots ,{秒}_{米}\in [0,+\infty )$. 特别是，我们可以从中推导出过程计数函数的联合概率。在这项工作中，我们获得了生成函数的大间隙渐近线，这是随着间隔大小的增长而渐近的。我们的结果适用于任意整数$米$, 在所有参数的情况下 ${秒}_1,\dots ,{秒}_{米}$，除了可能的一个，都是正的。这概括了两个已知结果：（1）Basor 和 Widom 的结果，对应于$米=1$ 和 ${秒}_1>0$, (2) 情况 $米=1$ 和 ${秒}_1=0$许多作者为此做出了贡献。我们还介绍了在正弦过程的细化和调节方面的一些应用。