  We analyze the phase structure of $SU(\infty)$ gauge theory at finite
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 temperature, a background field for the Polyakov loop, and a
quartic coupling, it exhibits a universal structure: in the large portion of
the parameter space, there is a continuous phase transition analogous to the
third-order phase transition of Gross, Witten and Wadia, but the order of phase
transition can be higher than third. We show that different confining
potentials give rise to drastically different behavior of the eigenvalue
density and the free energy. Therefore lattice simulations at large $N$ could
probe the order of phase transition and test our results.
