Introduction to the Matrix Integrals Method

Abstract

This chapter is devoted to a description of the intriguing connection between map enumeration and matrix integrals. This connection was first established in [143] for the purposes of matrix models of quantum gravity. It was later reinvented by mathematicians [138] in the computation of the Euler characteristic of moduli spaces of complex curves. Our presentation follows [25] and we show, along the lines of [138], how it leads to the enumeration of one-face maps. We also relate, following [87], [299], the universal one-matrix model to the Korteweg—de Vries (KdV) hierarchy of partial differential equations. The results of this chapter will be used in the next one in the description of the Harer—Zagier computation of the Euler characteristic of moduli spaces of curves, as well as in the study of Witten’s conjecture, which is now Kontsevich’s theorem.