Solving matrix models in the 1/N-expansion

Abstract

A description is given of properties of the most general multisupport solutions of one-matrix models, beginning with the one-matrix model in the presence of hard walls, that is, the case where the eigenvalue support is confined to several fixed intervals of the real axis. The eigenvalue model, which generalizes the one-matrix model to the Dyson gas case, is then considered. It is shown 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. The diagrammatic technique is developed for finding the correlation functions and the free energy of these models in all orders of the 'tHooft expansion in the reciprocal matrix size. In all cases these quantities can be formulated in terms of structures associated with the spectral curves.