General beta Jacobi corners process and the Gaussian Free Field

TL;DR

This paper establishes that the Gaussian Free Field models the large-scale fluctuations of a multilevel beta Jacobi ensemble, connecting random matrix theory with hypergeometric functions and exploring asymptotic behaviors.

Contribution

It introduces a novel connection between beta Jacobi ensembles and the Gaussian Free Field via Macdonald process degenerations, extending understanding of their asymptotics.

Findings

Gaussian Free Field describes global fluctuations

Connection between Jacobi ensembles and hypergeometric functions

Analysis of the beta to infinity limit

Abstract

We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.