Branched polymers from a double-scaling limit of matrix models

Multicritical potentials and correlation functions are given for models of rectangular M × N matrices, in the limit that N goes to infinity. These models are soluble without using orthogonal polynomials, and describe filamentary random surfaces, or, equivalently, a phase of branched polymers. It is shown that the equations describing multicritical behaviour are obtained from the hierarchy of flows that preserve Burgers' equation. Instanton solutions are studied - they imply that only the k = 2 model is unitary, and that the coefficients (for arbitrary k) of g_st^l is the perturbative expansion of the specific heat grow as l!.