Matrix Integrals and Feynman Diagrams in the Kontsevich Model

Abstract

We review some relations occurring between the combinatorial intersection theory on the moduli spaces of stable curves and the asymptotic behavior of the 't Hooft-Kontsevich matrix integrals. In particular, we give an alternative proof of the Witten-Di~Francesco-Itzykson-Zuber theorem ---which expresses derivatives of the partition function of intersection numbers as matrix integrals--- using techniques based on diagrammatic calculus and combinatorial relations among intersection numbers. These techniques extend to a more general interaction potential.