Effective theories for the thermal Wilson line are constructed in an $\mathrm{SU}\left(N\right)\mathrm{}$ gauge theory at nonzero temperature. I propose that the order of the deconfining phase transition for $Z\left(N\right)\mathrm{}$ Wilson lines is governed by the behavior of $\mathrm{SU}\left(N\right)\mathrm{}$ Wilson lines. In a mean field theory, the free energy in the deconfined phase is controlled by the condensate for $Z\left(N\right)\mathrm{}$ Wilson lines. Numerical simulations on the lattice, and the mean field theory for $Z\left(3\right)\mathrm{}$ Wilson lines, suggest that about any finite temperature transition in QCD the dominant correlation length increases by a large, uniform factor, of order five.

• Received 27 June 2000

DOI:https://doi.org/10.1103/PhysRevD.62.111501