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arXiv 2109.09292math.PR
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Bessel场的行列式结构

Determinantal structures for Bessel fields

Lucas Benigni, Pei-Ken Hung, Xuan Wu

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摘要：

贝塞尔场$\mathcal{B}=\{\mathcal{B}(α,t), α\in\mathbb{N}_0, t\in\mathbb{R}\}$是一个二元随机场，使得对于每个$(α,t)$，$\mathcal{B}(α,t)$具有指数为$α$的贝塞尔点过程的定律。贝塞尔场是拉盖尔场的硬边标度极限，是经典拉盖尔酉系综的自然扩张。最近在[LW 21]中证明了对于固定的$α$，$\{\mathcal{B}(α,t), t\in\mathbb{R}\}$是平方Bessel-Gibbsian线系综.在本文中，我们发现了丰富的可积结构的贝塞尔场：沿类时或类空路径，$\mathcal{B}$是一个具有显式相关核的决定点过程;对于固定的$t$，$\{\mathcal{B}(α,t),α\in\mathbb{N}_0\}$是一个指数吉布斯线系综。

英文摘要：

A Bessel field $\mathcal{B}=\{\mathcal{B}(α,t), α\in\mathbb{N}_0, t\in\mathbb{R}\}$ is a two-variable random field such that for every $(α,t)$, $\mathcal{B}(α,t)$ has the law of a Bessel point process with index $α$. The Bessel fields arise as hard edge scaling limits of the Laguerre field, a natural extension of the classical Laguerre unitary ensemble. It is recently proved in [LW21] that for fixed $α$, $\{\mathcal{B}(α,t), t\in\mathbb{R}\}$ is a squared Bessel Gibbsian line ensemble. In this paper, we discover rich integrable structures for the Bessel fields: along a time-like or a space-like path, $\mathcal{B}$ is a determinantal point process with an explicit correlation kernel; for fixed $t$, $\{\mathcal{B}(α,t),α\in\mathbb{N}_0\}$ is an exponential Gibbsian line ensemble.

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