  We study generalizations of the Gross--Witten--Wadia unitary matrix model for
the special orthogonal and symplectic groups. We show using a standard Coulomb
gas treatment -- employing a path integral formalism for the ungapped phase and
resolvent techniques for the gapped phase with one coupling constant -- that in
the large $N$ limit, the free energy normalized modulo the square of the gauge
group rank is twice the value for the unitary case. Using generalized Cauchy
identities for character polynomials, we then demonstrate the universality of
this phase transition for an arbitrary number of coupling constants by linking
this model to the random partition based on the Schur measure.
