Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles

TL;DR

This paper offers partial proofs for conjectures by Witte and Forrester on the combinatorial structure of Gaussian Beta Ensembles, utilizing topological recursion and proposing generalized conjectures for correlation functions.

Contribution

It introduces a simplified proof for certain coefficients and extends the conjectures to all correlation functions at coinciding points.

Findings

Partial proofs of Witte and Forrester's conjectures.

A generalized conjecture for all correlation functions.

Application of topological recursion at coinciding points.

Abstract

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $\beta$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion adapted for general $\beta$ Gaussian case. In particular, the paper uses a version at coinciding points that provides a simple proof for some of the coefficients involved in the conjecture. Additionally, we propose a generalized version of the conjectures for all correlation functions evaluated at coinciding points.