  We discuss how to extract renormalized from bare Polyakov loops in SU(N)
lattice gauge theories at nonzero temperature in four spacetime dimensions.
Single loops in an irreducible representation are multiplicatively renormalized
without mixing, through a renormalization constant which depends upon both
representation and temperature. The values of renormalized loops in the four
lowest representations of SU(3) were measured numerically on small, coarse
lattices. We find that in magnitude, condensates for the sextet and octet loops
are approximately the square of the triplet loop. This agrees with a large $N$
expansion, where factorization implies that the expectation values of loops in
adjoint and higher representations are just powers of fundamental and
anti-fundamental loops. For three colors, numerically the corrections to the
large $N$ relations are greatest for the sextet loop, $\leq 25%$; these
represent corrections of $\sim 1/N$ for N=3. The values of the renormalized
triplet loop can be described by an SU(3) matrix model, with an effective
action dominated by the triplet loop. In several ways, the deconfining phase
transition for N=3 appears to be like that in the $N=\infty$ matrix model of
Gross and Witten.
