Summing up perturbation series around superintegrable point

Abstract

We work out explicit formulas for correlators in the Gaussian matrix model perturbed by a logarithmic potential, i.e. by inserting Miwa variables. In this paper, we concentrate on the example of a single Miwa variable. The ordinary Gaussian model is superintegrable, i.e. the average of the Schur functions $S_Q$ is an explicit function of the Young diagram $Q$. The question is what happens to this property after perturbation. We show that the entire perturbation series can be nicely summed up into a kind of Borel transform of a universal exponential function, while the dependence on $R$ enters through a polynomial factor in front of this exponential. Moreover, these polynomials can be described explicitly through a single additional structure, which we call ``truncation'' of the Young diagram $Q$. It is unclear if one can call this an extended superintegrability, but at least it is a tremendously simple deformation of it. Moreover, the vanishing Gaussian correlators remain vanishing and, hence, are not deformed at all.

arXiv comment15 pages + Appendix (7 pages)

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• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• CMM formula as superintegrability property of unitary model
• Itzykson-Zuber correlators from character expansion
• Towards mixed phase correlators in monomial matrix models