  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.
