Introduction to Nonperturbative 2D Quantum Gravity

Abstract

We give an elementary introduction to the theory of matrix models, as applied to the study of 2 D quantum gravity. Starting from the sum over surfaces, we explain carefully the steps leading to the non-perturbative solution: discretization, duality, diagram-counting techniques, 1/N expansion, double-scaling limit, etc. Many examples are worked out in detail, and a pedagogical discussion of the concepts of universality and the non-perturbative ambiguity is presented.

Quasi-exactly solvable problems and random matrix theory

“…The method of orthogonal polynomials [1] is one of the basic tools for doing calculations in the theory of large random matrices (see e.g. recent review papers [2,3,4]). In order to remind the reader the idea of this method, let us consider for definiteness a model of random hermitean matrices Φ N of size (N + 1) × (N + 1) described by the potential V (Φ N ).…”