Work

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

Year: 2025

Type: article

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 to 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 $$\beta $$ β 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.

Source: The Ramanujan Journal

DOI: 10.1007/s11139-025-01218-z

Authors: Sung-Soo Byun, Peter J. Forrester

Institutions: Seoul National University, The University of Melbourne

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990

Funders: National Research Foundation, University of Melbourne, National Research Foundation of Korea, Australian Research Council