The Kontsevich matrix integral: convergence to the Painlevé hierarchy and Stokes' phenomenon

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 tronquée solution of the first Painlevé 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.

33 pages, 9 figures; ver2 improved introduction

References in corpus (1)

• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$

Cited by in corpus (4)

• Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials
• The Kontsevich-Penner matrix integral, isomonodromic tau functions and open intersection numbers
• Matrix models for stationary Gromov-Witten invariants of the Riemann sphere
• Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles