We show that the Kontsevich integral on matrices () is the isomonodromic tau function associated to a Riemann-Hilbert Problem. The approach allows us to gain control of the analysis of the convergence as . 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,e solution of the first Painlev, 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.

The Kontsevich Matrix Integral: Convergence to the Painlevé Hierarchy and Stokes’ Phenomenon / Bertola, M.; Cafasso, M.. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 352:2(2017), pp. 585-619. [10.1007/s00220-017-2856-3]

The Kontsevich Matrix Integral: Convergence to the Painlevé Hierarchy and Stokes’ Phenomenon

Bertola, M.;Cafasso, M.
2017-01-01

Abstract

We show that the Kontsevich integral on matrices () is the isomonodromic tau function associated to a Riemann-Hilbert Problem. The approach allows us to gain control of the analysis of the convergence as . 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,e solution of the first Painlev, 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.
2017
352
2
585
619
https://arxiv.org/abs/1603.06420
https://link.springer.com/article/10.1007%2Fs00220-017-2856-3
Bertola, M.; Cafasso, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/50129
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