Interlacing adjacent levels of $\beta$--Jacobi corners processes

TL;DR

This paper investigates the fluctuations between adjacent levels in the $eta$--Jacobi corners process, revealing their connection to the derivative of the 2D Gaussian Free Field using advanced algebraic tools.

Contribution

It introduces a novel analysis of the asymptotic fluctuations in the $eta$--Jacobi corners process and links them to Gaussian Free Field derivatives.

Findings

Fluctuations converge to the derivative of the 2D Gaussian Free Field.

Provides integral formulas for difference operators in the process.

Extends classical Jacobi ensemble results to a multilevel $eta$-extension.

Abstract

We study the asymptotics of the global fluctuations for the difference between two adjacent levels in the $\beta$--Jacobi corners process (multilevel and general $\beta$ extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the $2d$ Gaussian Free Field. Our main tools are integral forms for the (Macdonald-type) difference operators originating from the shuffle algebra.