On the hard edge limit of the zero temperature Laguerre beta corners process

TL;DR

This paper investigates the hard edge limit of a multilevel Laguerre beta-ensemble at zero temperature, revealing Gaussian asymptotics with covariance linked to Fourier-Bessel series and connections to additive polymers.

Contribution

It introduces a novel analysis of the hard edge limit using dual polynomials and Fourier-Bessel series, extending previous methods to this specific ensemble.

Findings

Gaussian asymptotics at the hard edge

Covariance matrix expressed via Fourier-Bessel series

Representation as partition functions of additive polymers

Abstract

We study the hard edge limit of a multilevel extension of the Laguerre $\beta$-ensemble at zero temperature. In particular, we show that asymptotically the ensemble is given by Gaussians with covariance matrix expressible in terms of the Fourier-Bessel series. These Gaussians also have an explicit representation as the partition functions of additive polymers arising from a random walk on roots of the Bessel functions. Our approach builds off of the one introduced in arxiv:2009.02006 and is rooted in using the theory of dual and associated polynomials to diagonalize transition matrices relating levels of the ensemble.