Summary: From the text: We study the double scaling limit of mKdV type, realized in the two-cut Hermitian matrix model. Building on the work of Periwal and Shevitz and of Nappi, we find an exact solution including all odd scaling operators, in terms of a hierarchy of flows of \(2\times 2\) matrices. We derive from it loop equations which can be expressed as Virasoro constraints on the partition function. We discover a ‘pure topological’ phase of the theory in which all correlation functions are determined by recursion relations. We also examine macroscopic loop amplitudes, which suggest a relation to \(2\)D gravity coupled to dense polymers. Since we cannot deduce the continuum theory directly from the lattice we are forced to compare the structure of the exact solution of the lattice theory with the structure of known continuum theories. With this motivation, we describe below the exact solution of the lattice theory in some detail. In Section 2 we derive from the lattice the complete string and flow equations for two-cut models, including all even and odd perturbations in the potential. In Section 3 we show how these equations can be rewritten as Virasoro-type constraints. This reformulation suggests the existence of a topological phase of the theory, discussed in Section 4. In Section 5 we make some brief remarks about macroscopic loops in these theories. In the conclusions we attempt to draw some inferences about what the continuum formulation of these models might be.