Deformation of superintegrability in the Miwa-deformed Gaussian matrix model

Abstract

We consider an arbitrary deformation of the Gaussian matrix model parametrized by Miwa variables za. 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 polynomial made from the values of skew Schur functions at the Gaussian locus pk=δk,2. This linear operation includes multiplication with an exponential eza2/2 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 superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in great detail the generating function of correlators ⟨(TrX)k⟩ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian β-ensemble. Published by the American Physical Society 2024