  We propose a phenomenological matrix model to study string theory in AdS_5
\times S_5 in the canonical ensemble. The model reproduces all the known
qualitative features of the theory. In particular, it gives a simple effective
potential description of Euclidean black hole nucleation and the tunnelling
between thermal AdS and the big black hole. It also has some interesting
predictions. We find that there exists a critical temperature at which the
Euclidean small black hole undergoes a Gross-Witten phase transition. We
identify the phase transition with the Horowitz-Polchinski point where the
black hole horizon size becomes comparable to the string scale. The appearance
of the Hagedorn divergence of thermal AdS is due to the merger of saddle points
corresponding to the Euclidean small black hole and thermal AdS. The merger can
be described in terms of a cusp (A_3) catastrophe and divergences at the
perturbative string level are smoothed out at finite string coupling using
standard techniques of catastrophe theory.
