The Kontsevich matrix integral: convergence to the Painlev\'e hierarchy and Stokes' phenomenon

TL;DR

This paper demonstrates that the Kontsevich matrix integral converges to the isomonodromic tau function of a Painlevé hierarchy solution, linking matrix models, Riemann-Hilbert problems, and Stokes' phenomena.

Contribution

It establishes the connection between the Kontsevich integral and Painlevé hierarchy tau functions, providing a new analytic approach to study their convergence and Stokes' phenomena.

Findings

Kontsevich integral is the isomonodromic tau function of a Riemann-Hilbert problem.

Limit of the integral yields a special Painlevé hierarchy solution.

Multiple tau functions are analytic in parameter sectors, with asymptotic expansions related to the Witten-Kontsevich tau function.

Abstract

We show that the Kontsevich integral on $n\times n$ matrices ($n< \infty$) is the isomonodromic tau function associated to a $2\times 2$ Riemann--Hilbert problem. The approach allows us to gain control of the analysis of the convergence as $n\to\infty$. By an appropriate choice of the external source matrix in Kontsevich's integral, we show that the limit produces the isomonodromic tau function of a special tronqu\'ee solution of the first Painlev\'e hierarchy, and we identify the solution in terms of the Stokes' data of the associated linear problem. We also show that there are several tau functions that are analytic in appropriate sectors of the space of parameters and that the formal Witten-Kontsevich tau function is the asymptotic expansion of each of them in their respective sectors, thus providing an analytic tool to analyze its nonlinear Stokes' phenomenon.