  In this paper we derive a hierarchy of differential equations which uniquely
determine the coefficients in the asymptotic expansion, for large $N$, of the
logarithm of the partition function of $N \times N$ Hermitian random matrices.
These coefficients are generating functions for graphical enumeration on
Riemann surfaces. The case that we particularly consider is for an underlying
measure that differs from the Gaussian weight by a single monomial term of
degree $2\nu$. The coupling parameter for this term plays the role of the
independent dynamical variable in the differential equations. From these
equations one may deduce functional analytic characterizations of the
coefficients in the asymptotic expansion. Moreover, this ode system can be
solved recursively to explicitly construct these coefficients as functions of
the coupling parameter. This analysis of the fine structure of the asymptotic
coefficients can be extended to multiple coupling parameters and we present a
limited illustration of this for the case of two parameters.
