The Bessel Line Ensemble

TL;DR

This paper constructs the Bessel line ensemble, a collection of continuous random curves with stationarity, specific finite-dimensional distributions, and a novel resampling invariance, derived from the hard edge scaling limit of squared Bessel processes.

Contribution

It introduces the Bessel line ensemble with its stationarity, resampling invariance, and connection to the Dyson Bessel process, expanding the understanding of Bessel-related stochastic processes.

Findings

The Bessel line ensemble is stationary under horizontal shifts.

It exhibits a novel resampling invariance with respect to non-intersecting squared Bessel bridges.

Finite-dimensional distributions are given by the extended Bessel kernel.

Abstract

In this paper, we construct the Bessel line ensemble, a countable collection of continuous random curves. This line ensemble is stationary under horizontal shifts with the Bessel point process as its one-time marginal. Its finite dimensional distributions are given by the extended Bessel kernel. Furthermore, it enjoys a novel resampling invariance with respect to non-intersecting squared Bessel bridges. The Bessel line ensemble is constructed by extracting the hard edge scaling limit of a collection of independent squared Bessel processes starting at the origin and never being conditioned to intersect. This process is also known as the Dyson Bessel process, and it arises as the evolution of the eigenvalues of the Laguerre unitary ensemble with i.i.d. complex Brownian entries.