  We study the deconfining phase transition in SU(N) gauge theories at nonzero
temperature using a matrix model of Polyakov loops. The most general effective
action, including all terms up to two spatial derivatives, is presented. At
large N, the action is dominated by the loop potential: following Aharony et
al., we show how the Gross--Witten model represents an ultra-critical point in
this potential. Although masses vanish at the Gross--Witten point, the
transition is of first order, as the fundamental loop jumps only halfway to its
perturbative value. Comparing numerical analysis of the N=3 matrix model to
lattice simulations, for three colors the deconfining transition appears to be
near the Gross--Witten point. To see if this persists for N >= 4, we suggest
measuring within a window ~1/N^2 of the transition temperature.
