Scaling and Universality in Random Matrix Models

Partition functions of random matrix models provide generating functions for a number of combinatorial and physical problems: Enumeration of graphs on Riemannian surfaces and quantum gravity, models of statistical mechanics on random surfaces, enumeration of knots and links, meanders, and others. Critical points and double scaling limits of random matrix models determine in this context large N asymptotics of the quantities under consideration. In this introductory series of 4 lectures we will discuss ensembles of random matrix models, integrable structures for correlation functions of eigenvalues of random matrices and their relation to orthogonal polynomials. We will consider the Riemann-Hilbert approach to semiclassical asymptotics of orthogonal polynomials, the Deift-Zhou nonlinear steepest descent method, and scaling limits and universality of the eigenvalue correlation functions. Finally, we will discuss large N asymptotics of the free energy of the ensemble of random matrices, critical asymptotics and double scaling limits at critical points.