In the general framework of one-Hermitian-matrix models, we study critical behaviors such that δx~δRm/n~δSm; δx, δS and δR are, respectively, the bare cosmological constant and the orthogonal-polynomial coefficients around criticality. On the sphere, we prove the existence of consistent multicriticality conditions such that string equations exhibit the above behavior. We define a double scaling limit and write down exact equations for the specific heat for any (m, n) model. Their solutions are unambiguous and the only corrections come from genus-one topology. We compute exact correlation functions for well-defined scaling operators. These belong to two different sectors. One of them is such that any squared operator vanishes when inserted in any correlation function. We discuss briefly the flows between these multicritical points as well as the nature of the 2D field theories coupled to gravity which they can describe.
Genus expansion of matrix models and ћ expansion of KP hierarchy