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We show that the Kontsevich integral on $n\times n$ matrices ($n< \infty$) isthe isomonodromic tau function associated to a $2\times 2$ Riemann--Hilbertproblem. The approach allows us to gain control of the analysis of theconvergence as $n\to\infty$. By an appropriate choice of the external sourcematrix in Kontsevich's integral, we show that the limit produces theisomonodromic tau function of a special tronqu\'ee solution of the firstPainlev\'e hierarchy, and we identify the solution in terms of the Stokes' dataof the associated linear problem. We also show that there are several taufunctions that are analytic in appropriate sectors of the space of parametersand that the formal Witten-Kontsevich tau function is the asymptotic expansionof each of them in their respective sectors, thus providing an analytic tool toanalyze its nonlinear Stokes' phenomenon.

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