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Recently lattice simulation in pure Yang-Mills theory exposes significantquadratic corrections for both the thermodynamic quantities and therenormalized Polyakov loop in the deconfined phase. These terms are previouslyfound to appear naturally for ${\mathcal N}=4$ Super Yang-Mills~(SYM) on asphere at strong coupling, through the gauge/gravity duality. Here we extendthe investigation to the weak coupling regime, and for general large-$N$ gaugetheories. Employing the matrix model description, we find some novel behaviorin the deconfined phase, which is not noticed in the literature. Due to thenon-uniform eigenvalue distribution of the holonomy around the time circle, thedeviation 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 differenttheories. This statement is also true when perturbative corrections to thesingle-particle partition functions are included. The corrections to thePolyakov loop and higher moments of the distribution function combine to give auniversal term, $T/4$, in the free energy. These differences between the weakand strong coupling regime could be easily explained if a strong/weak couplingphase transition occurs in the deconfined phase of large-$N$ gauge theories ona compact manifold.

Thermodynamics of large-$N$ gauge theories on a sphere: weak versus strong coupling