Solving matrix models in $1/N$-expansion

We describe properties of most general multisupport solutions to one-matrix models. We begin with the one-matrix model in the presence of hard walls, i.e., in the case where the eigenvalue support is confined to several fixed intervals of the real axis. We then consider the eigenvalue model, which generalizes the one-matrix model to the Dyson gas case. We show that in all these cases, the structure of the solution at the leading order is described by semiclassical, or generalized Whitham–Krichever hierarchies. Derivatives of tau-functions for these solutions are associated with families of Riemann surfaces (spectral curves with possible double points) and satisfy the Witten–Dijkgraaf–Verlinde–Verlinde equations. We develop the diagrammatic technique for finding correlation functions and free energy of these models in all orders of the 't Hooft expansion in the reciprocal matrix size. In all cases, these quantities can be formulated in terms of strucutures associated with the spectral curves.