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<p>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 (<i>q</i>,&#xa0;<i>t</i>) generalisation of the Gaussian <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1218_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月22日

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  • Sung-Soo Byun,
  • Peter J. Forrester

摘要

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.