Jacobi Beta Ensemble and $b$-Hurwitz Numbers

TL;DR

This paper links Jacobi beta ensemble correlators to b-Hurwitz numbers, providing new insights into their structure and connections to Hurwitz maps, with implications for integrals and limiting cases.

Contribution

It introduces a novel expression of Jacobi beta ensemble correlators using b-Hurwitz numbers, expanding the understanding of their combinatorial and integral representations.

Findings

Correlators expressed via b-Hurwitz numbers

Laguerre limit analyzed

Connections to colored monotone Hurwitz maps established

Abstract

We express correlators of the Jacobi $\beta$ ensemble in terms of (a special case of) $b$-Hurwitz numbers, a deformation of Hurwitz numbers recently introduced by Chapuy and Dolega. The proof relies on Kadell's generalization of the Selberg integral. The Laguerre limit is also considered. All the relevant $b$-Hurwitz numbers are interpreted (following Bonzom, Chapuy, and Dolega) in terms of colored monotone Hurwitz maps.