Large N expansions and Painlev\'e hierarchies in the Hermitian matrix model

We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlev\'e I and Painlev\'e II hierarchies associated with one-cut and two-cut critical models respectively.