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Discrete Painlevé equations and their appearance in quantum gravity. (English) Zbl 0742.35047
Summary: We discuss an algorithmic approach for both deriving discrete analogues of Painlevé equations as well as using such equations to characterize “similarity” reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painlevé I can be used to characterize “similarity” solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painlevé IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painlevé I in terms of the solution of discrete Painlevé I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure $\exp[- t\sb 1z\sp 2-t\sb 2z\sp 4-t\sb 3z\sp 6]$.

35Q40PDEs in connection with quantum mechanics
35Q53KdV-like (Korteweg-de Vries) equations
Full Text: DOI
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