  We study phase transitions in $SU(\infty)$ gauge theories at nonzero
temperature using matrix models. Our basic assumption is that the effective
potential is dominated by double trace terms for the Polyakov loops. As a
function of the various parameters, related to terms linear, quadratic, and
quartic in the Polyakov loop, the phase diagram exhibits a universal structure.
In a large region of this parameter space, there is a continuous phase
transition whose order is larger than second. This is a generalization of the
phase transition of Gross, Witten, and Wadia (GWW). Depending upon the detailed
form of the matrix model, the eigenvalue density and the behavior of the
specific heat near the transition differ drastically. We speculate that in the
pure gauge theory, that although the deconfining transition is
thermodynamically of first order, it can be nevertheless conformally symmetric
at infinite $N$.
