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arXiv · 2505.12927

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

Sung-Soo Byun·Peter J. Forrester
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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.

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Sung-Soo Byun, Peter J. Forrester. 2025-05-19. On the superintegrability of the Gaussian $\beta$ ensemble and its $(q,t)$ generalisation. https://arxiv.org/abs/2505.12927

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