Large $N$ Phase Transition In The Heat Kernel On The $U(N)$ Group
  The large N phase transition point is investigated in the heat kernel on the $U(N)$ group with respect to arbitrary boundary conditions. A simple functional relation is found relating the density of eigenvalues of the boundary field to the saddle point shape of the typical Young tableaux in the large $N$ limit of the character expansion of the heat kernel. Both strong coupling and weak coupling phases are investigated for some particular cases of the boundary holonomy. 
Vladimir A. Kazakov and Thomas Wynter