Determinantal structures for Bessel fields

TL;DR

This paper explores the integrable structures of Bessel fields, showing they form determinantal point processes along certain paths and are Gibbsian line ensembles, revealing deep connections to random matrix theory and stochastic processes.

Contribution

It uncovers explicit determinantal structures and Gibbsian properties of Bessel fields, extending understanding of their integrable nature and connections to random matrix ensembles.

Findings

Bessel fields are determinantal point processes along time-like and space-like paths.

For fixed time, Bessel fields form exponential Gibbsian line ensembles.

The work links Bessel fields to integrable probability and random matrix theory.

Abstract

A Bessel field $\mathcal{B}=\{\mathcal{B}(\alpha,t), \alpha\in\mathbb{N}_0, t\in\mathbb{R}\}$ is a two-variable random field such that for every $(\alpha,t)$, $\mathcal{B}(\alpha,t)$ has the law of a Bessel point process with index $\alpha$. 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 $\alpha$, $\{\mathcal{B}(\alpha,t), t\in\mathbb{R}\}$ 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 $t$, $\{\mathcal{B}(\alpha,t),\alpha\in\mathbb{N}_0\}$ is an exponential Gibbsian line ensemble.