Abstract
We consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp\(\sum\limits_{j = 1}^N {t_{j^{Z^{2j,} } } N \geqq 3} \). We show that after the appropriate modification of the contour of integration the Cross-Migdal-Douglas-Shenker limit to the Painlevé I equation (in the generic case of the pure gravity) is valid and calculate the nonperturbative parameters of the corresponding Painlevé function. Our approach is based on the WKB-analysis of the L-A pair corresponding to the discrete string equation in the framework of the Inverse Monodromy Method. Here we extend our results, which were obtained before for the particular casesN=2,3. Our analysis complements the isomonodromy approach proposed by G. Moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself.
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Communicated by N. Yu. Reshetikhin
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Fokas, A.S., Its, A.R. & Kitaev, A.V. The isomonodromy approach to matric models in 2D quantum gravity. Commun.Math. Phys. 147, 395–430 (1992). https://doi.org/10.1007/BF02096594
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DOI: https://doi.org/10.1007/BF02096594