  The dynamics of finite temperature U(N) gauge theories on $S^3$ can be
described, at weak coupling, by an effective unitary matrix model. Here we
present an exact solution to these models, for any value of $N$, in terms of a
sum over representations. Taking the large $N$ limit of this solution provides
a new perspective on the deconfinement transition which is supposed to be dual
to the Hawking-Page transition. The large $N$ phase transition manifests itself
here in a manner similar to the Douglas-Kazakov phase transition in 2d
Yang-Mills theory. We carry out a complete analysis of the saddle
representation in the simplest case involving only the order parameter ${\rm
Tr}U$. We find that the saddle points corresponding to thermal $AdS$, the small
black hole and the large black hole can all be described in terms of free
fermions. They all admit a simple phase space description {\it a la} the BPS
geometries of Lin, Lunin and Maldacena.
