Combinatorics and large genus asymptotics of the Br\'ezin--Gross--Witten numbers

TL;DR

This paper investigates the combinatorial and asymptotic properties of Brézin–Gross–Witten numbers, revealing their large genus behavior, polynomiality, and proposing new conjectures with applications to integrable hierarchies.

Contribution

It provides the first proof of the uniform large genus asymptotics of normalized BGW numbers and introduces new conjectures supported by extensive numerical data.

Findings

Established the large genus asymptotics of BGW numbers

Proved the polynomiality phenomenon in large genus limits

Proposed new conjectures on monotonicity and integrality of BGW numbers

Abstract

In this paper, we study combinatorial and asymptotic properties of some interesting rational numbers called the Br\'ezin--Gross--Witten (BGW) numbers, which can be represented as the intersection numbers of psi and Theta classes on the moduli space of stable algebraic curves. In particular, we discover and prove the uniform large genus leading asymptotics of certain normalized BGW numbers, and give a new proof of the polynomiality phenomenon for the large genus asymptotics. We also propose, with extensive numerical data, several new conjectures including monotonicity and integrality on the BGW numbers. Applications to the Painlev\'e II hierarchy and to the BGW-kappa numbers are given.