On the superintegrability of the Gaussian $β$ ensemble and its $(q,t)$ generalisation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916743832666112 |
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| author | Byun, Sung-Soo Forrester, Peter J. |
| author_facet | Byun, Sung-Soo Forrester, Peter J. |
| contents | 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 $β$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the superintegrability of the Gaussian $β$ ensemble and its $(q,t)$ generalisation Byun, Sung-Soo Forrester, Peter J. Mathematical Physics 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 $β$ 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. |
| title | On the superintegrability of the Gaussian $β$ ensemble and its $(q,t)$ generalisation |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2505.12927 |