Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

Abstract

We consider an arbitrary deformation of the Gaussian matrix model parameterized by Miwa variables $z_a$. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a {\it finite degree} polynomial made from the values of skew Schur functions at the Gaussian locus $p_k={\delta }_{k,2}$. This linear operation includes multiplication with an exponential $e^{z_a^2/2}$ and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary $K$-polynomials, which appeared in {\it bilinear} superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in the very detail the generating function of correlators $<(\Tr X)^k>$ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian $\beta$-ensemble.

21 pages. arXiv admin note: text overlap with arXiv:2401.14392

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• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Macdonald deformation of Vogel's universality and link hyperpolynomials
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