On the superintegrability of the Gaussian $\beta$ ensemble and its $(q,t)$ generalisation

TL;DR

This paper proves superintegrability for a $(q,t)$ generalisation of the Gaussian $eta$ ensemble using Macdonald polynomial theory, leading to duality and functional equations for spectral moments.

Contribution

It introduces a superintegrability identity for the $(q,t)$ Gaussian $eta$ ensemble based on multivariable Al-Salam and Carlitz polynomials, extending previous results.

Findings

Proves superintegrability for the $(q,t)$ Gaussian $eta$ ensemble.

Derives a duality formula for ensemble averages.

Establishes a functional equation for spectral moments.

Abstract

In the present context, superintegrability is a property of certain probability density functions coming from matrix models, which relates to the average over a distinguished basis of symmetric functions, typically the Jack or Macdonald polynomials. It states that the average can be computed according a certain combination of those same polynomials, now specialised by specific substitutions when expressed in terms of the power sum basis. For a particular $(q,t)$ generalisation of the Gaussian $\beta$ ensemble from random matrix theory, known independently from the consideration of certain integrable gauge theories, we use results developed in a theory of multivariable Al-Salam and Carlitz polynomials based on Macdonald polynomials to prove the superintegrability identity. This then is used to deduce a duality formula for these same averages, which in turn allows for a derivation of a functional equation for the spectral moments.