The Kontsevich Matrix Integral: Convergence to the Painlevé Hierarchy and Stokes' Phenomenon
Abstract
We show that the Kontsevich integral on {n× n} matrices ({n < ∞}) is the isomonodromic tau function associated to a {2× 2} RiemannHilbert Problem. The approach allows us to gain control of the analysis of the convergence as {n\to∞}. 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 WittenKontsevich 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2017
 DOI:
 10.1007/s0022001728563
 arXiv:
 arXiv:1603.06420
 Bibcode:
 2017CMaPh.352..585B
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 34M40;
 34M55;
 34M56;
 35Q15
 EPrint:
 33 pages, 9 figures