Matrix models of two-dimensional quantum gravity and isomonodromy solutions of “discrete Painleve equations”

Method for investigation of double scaling limits in two-dimensional string models of quantum gravity is formulated. In fact, ot is shown that the study of such limits reduces to the isomonodromy deformation method for integrable discrete equations. The connection between “universality” and isomonodromy properties of a model is found. The model $\Phi^4$ is considered in details. The partition function of the model appeared to be $\tau$-function for the fourth Painlevé equation $\mathbb{P}_4$ and Kac–Moerbeke lattice. The properties of Bäcklund transformations for $\mathbb{P}_4$ are studied in details.