A quick derivation of the loop equations for random matrices

The loop equations of random matrix theory are a hierarchy of equations born of attempts to obtain explicit formulae for generating functions of map enumeration problems. These equations, originating in the physics of 2-dimensional quantum gravity, have lacked mathematical justification. The goal of this paper is to provide a complete and short proof, relying on a recently established complete asymptotic expansion for the random matrix theory partition function. 1. Background and preliminaries The study of the unitary ensembles (UE) of random matrices [Mehta 1991], begins with a family of probability measures on the space of N N Hermitian matrices. The measures are of the form d t D 1 z ZN exp f N Tr ŒVt.M /gdM; where the function Vt is a scalar function, referred to as the potential of the external field, or simply the “external field” for short. Typically it is taken to be a polynomial, and written as follows: Vt D V . I t1; : : : ; t /D 1 2 2 C X