  Recently lattice simulation in pure Yang-Mills theory exposes significant
quadratic corrections for both the thermodynamic quantities and the
renormalized Polyakov loop in the deconfined phase. These terms are previously
found to appear naturally for ${\mathcal N}=4$ Super Yang-Mills~(SYM) on a
sphere at strong coupling, through the gauge/gravity duality. Here we extend
the investigation to the weak coupling regime, and for general large-$N$ gauge
theories. Employing the matrix model description, we find some novel behavior
in the deconfined phase, which is not noticed in the literature. Due to the
non-uniform eigenvalue distribution of the holonomy around the time circle, the
deviation of the Polyakov loop from one starts from $1/T^3$ instead of $1/T^2$.
Such a power is fixed by the space dimension and do not change with different
theories. This statement is also true when perturbative corrections to the
single-particle partition functions are included. The corrections to the
Polyakov loop and higher moments of the distribution function combine to give a
universal term, $T/4$, in the free energy. These differences between the weak
and strong coupling regime could be easily explained if a strong/weak coupling
phase transition occurs in the deconfined phase of large-$N$ gauge theories on
a compact manifold.
