We consider unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. For the case of a quartic potential Bleher and Its obtained the double scaling limits of the eigenvalue correlation kernels at such a critical point. The limiting kernels are constructed out of functions associated with the second Painleve equation. We extend this result to general potentials.
(Based on joint work with Tom Claeys.)