  Recent work on the spectrum of the Euclidean Dirac operator spectrum show
that the exact microscopic spectral density can be computed in both random
matrix theory, and directly from field theory. Exact relations to effective
Lagrangians with additional quark species form the bridge between the two
formulations. Taken together with explicit computations in the chGUE random
matrix ensemble, a series of universality theorems are used to prove that the
finite-volume QCD partition function coincides exactly with the universal
double-microscopic limit of chUE random matrix partition functions. In the
limit where N_f and N_c both go to infinity with the ratio N_f/N_c fixed, the
relevant effective Lagrangian undergoes a third order phase transition of
Gross-Witten type.
