  A hermitian one-matrix model with an even quartic potential exhibits a
third-order phase transition when the cuts of the matrix model curve coalesce.
We use the known solutions of this matrix model to compute effective
superpotentials of an N=1, SU(N) supersymmetric Yang-Mills theory coupled to an
adjoint superfield, following the techniques developed by Dijkgraaf and Vafa.
These solutions automatically satisfy the quantum tracelessness condition and
describe a breaking to SU(N/2) x SU(N/2) x U(1). We show that the value of the
effective superpotential is smooth at the transition point, and that the
two-cut (broken) phase is more favored than the one-cut (unbroken) phase below
the critical scale. The U(1) coupling constant diverges due to the massless
monopole, thereby demonstrating Ferrari's general formula. We also briefly
discuss the implication of the Painleve II equation arising in the double
scaling limit.

High Energy Physics - Theory