Abstract
The goal of this article is to prove that the determinantal formulas of the Painlev'e 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary non-zero monodromy parameter. The result is established for two different Lax pairs associated to the Painlev'e 2 system, namely the Jimbo-Miwa Lax pair and the Harnad-Tracy-Widom Lax pair, whose spectral curves are not connected by any symplectic transformation. We provide a new method to prove the topological type property without using the insertion operators. In the process, taking the time parameter t to infinity gives that the symplectic invariants F(g) computed from the Hermite-Weber curve and the Bessel curve are equal to respectively. This result generalizes similar results obtained from random matrix theory in the special case where {\theta} = 0. We believe that this approach should apply for all 6 Painlev'e equations with arbitrary monodromy parameters. Explicit computations up to g = 3 are provided along the paper as an illustration of the results.
Highlights
In the past decade, the connection between random matrix theory, topological recursion and integrable systems has been developed intensively
In particular the main interest of the topological recursion is the definition of a series of numbers F (g) known as “symplectic invariants” that are invariant under a certain class of symplectic transformations of the initial spectral curve and that reconstruct the logarithm of the partition function when the spectral curve arises from a matrix model
PAINLEVE 2, TOPOLOGICAL RECURSION AND DETERMINANTAL FORMULAS 3 to associate a natural spectral curve to any 2 × 2 Lax pair and provided some determinantal formulas attached to the Lax pair. These determinantal formulas match the correlation functions and symplectic invariants obtained from the computation of the topological recursion on the spectral curve when some additional conditions, known generically as the “topological type” (TT) property, are satisfied
Summary
The connection between random matrix theory, topological recursion and integrable systems has been developed intensively. These determinantal formulas match the correlation functions and symplectic invariants obtained from the computation of the topological recursion on the spectral curve when some additional conditions, known generically as the “topological type” (TT) property, are satisfied These notions were extended successfully to n × n Lax pairs by Bergere, Borot and Eynard [2]. After presenting the topological recursion and the determinantal formulas, we will prove the TT property by proving the three conditions proposed in [3] This result proves that the generating functions for both sets of symplectic invariants FJ(Mg)(t) and FH(gT)W(t) defined from the spectral curves of JM pair and HTW pair give the corresponding tau-functions of Painleve 2 We show that the above discrepancy between F (g)’s is consistent with the integration constant computed in [18] (see Appendix E)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
