Graphic Enumerations and Discrete Painlevé Equations via Random Matrix Models

Abstract: We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients of orthogonal polynomials associated with the quartic matrix model satisfy the discrete type I Painlev\'e equation. Through the use of generating function techniques, we show that the planar contribution to the free energy is controlled by the Catalan numbers. We also develop a new systematic scheme of calculating higher-genus contributions to the topological expansion of the free energy of matrix models. It is important that our exact solutions are valid for finite-$N$ matrix models and no continuous limits are taken within our approach. To show the advantages of our approach, we provide new results of the topological expansion of the free energy for the finite-$N$ cubic matrix model.