The external field problem in the large N limit of QCD

Several problems in lattice gauge theories such as mean field theory or the few plaquette problem lead to the evaluation of the properties of one link in an external matrix source. This problem is solved here in the large N limit. There are two phases characterized by a single parameter, the average value of the inverse of the modulus of the eigenvalues of the external source. The third derivative of the free energy is discontinuous at the transition point.