Large gap asymptotics for the generating function of the sine point process

Abstract

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 \begin{equation*} \sum_{k_{1},...,k_{m} \geq 0} \mathbb{P}\Bigg(\bigcap_{j=1}^{m} \#\{\mbox{points in j-th interval}\}=k_{j}\Bigg)\prod_{j=1}^{m} s_{j}^{k_{j}}, \end{equation*} where $s_1,\dots ,s_m\in [0,1]$. 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.

51 pages, 9 figures. arXiv admin note: text overlap with arXiv:1812.02188

References in corpus (5)

• Randomly incomplete spectra and intermediate statistics
• Bulk Universality and Related Properties of Hermitian Matrix Models
• On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I
• Large gap asymptotics for Airy kernel determinants with discontinuities
• The sine process under the influence of a varying potential

Cited by in corpus (5)

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• Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition