Random supermatrices and critical behavior

The zero-dimensional MxM random matrix model with N=2 supersymmetry is presented using a manifest hermitian supermatrix formalism. The general expression for the partition function of such models is derived and analyzed. It is shown that when the superpotential satisfies a particular nonlinear ordinary differential equation, for which the solution is obtained, these models reduce to precisely the Penner matrix model, which is known to be related to c=1 bosonic string theory. The model is completely solved in the exact double-scaling limit in the case of the cubic superpotential, the critical behavior of which is shown to lie in the same universality class as that of the Penner matrix model. It is argued that, unlike the bosonic case, the solutions of the string equations of the random supermatrix model lie in different universality classes for the cubic and quartic k=2 potentials. (orig.)