Painlevé 2 Equation with Arbitrary Monodromy Parameter, Topological Recursion and Determinantal Formulas

The goal of this article is to prove that the determinantal formulas of the Painlevé 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary nonzero monodromy parameter. The result is established for a WKB expansion of two different Lax pairs associated with the Painlevé 2 system, namely the Jimbo–Miwa Lax pair and the Harnad–Tracy–Widom Lax pair, where a small parameter ħ is introduced by a proper rescaling. The proof is based on showing that these systems satisfy the topological type property introduced in Bergère et al. (Ann Henri Poincaré 16:2713, 2015 ), Bergère and Eynard ( arxiv:0901.3273 , 2009 ). In the process, we explain why the insertion operator method traditionally used to prove the topological type property is currently incomplete and we propose new methods to bypass the issue. Our work generalizes similar results obtained from random matrix theory in the special case of vanishing monodromies (Borot and Eynard in arXiv:1011.1418 , 2010 ; arXiv:1012.2752 , 2010 ). Explicit computations up to g = 3 are provided along the paper as an illustration of the results. Eventually, taking the time parameter t to infinity we observe that the symplectic invariants F ( g ) of the Jimbo–Miwa and Harnad–Tracy–Widom spectral curves converge to the Euler characteristic of moduli space of genus g Riemann surfaces.