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The isomonodromy approach to matrix models in 2D quantum gravity. (English) Zbl 0760.35051
The analysis performed in this article complements the isomonodromy approach, proposed by G. Moore [NATO ASI Ser., Ser. B 262, 157-190 (1991)], to the general string equations that come from the matrix model in the continuous limit and is original by the fact that the isomonodromy technique is applied to investigate the double-scaling limit itself. Based on the WKB-analysis of the $$L-A$$ pairs corresponding to the discrete string equation, the authors consider the double-scaling limit in the Hermitian matrix model for 2D quantum gravity associated with the measure $$\exp(t_ jz^{2j})$$ for $$N3$$, concretely show that the Cross- Migdal-Douglas-Shenker limit to the Painlevé I equation is valid after an appropriate modification of the contour of integration and calculate the nonperturbative parameters of the corresponding Painlevé function.
Reviewer: C.Dariescu (Iaşi)

##### MSC:
 35Q75 PDEs in connection with relativity and gravitational theory 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 83C45 Quantization of the gravitational field 39A12 Discrete version of topics in analysis
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