Fundamental modular region, Boltzmann factor, and area law in Lattice gauge theory

The thermodynamic limit of lattice gauge theory is derived in a gauge which is optimized to make all link variables as close to unity as possible. The derivation rests upon (1) a precise bound on the fundamental modular region and (2) a direct evaluation of the functional integral of the Wilson lattice by the saddle-point method that is valid in the thermodynamic limit. The result confirms the calculational scheme obtained previously, which differs from the Faddeev-Popov scheme by the incorporation of non-perturbative effects, but which remains perturbatively renormalizable. The lattce Faddeev-Popov propagator, which appears in the modified action, acquires a dipole singularity at zero momentum, characteristic of long-range correlations. This is sufficient to produce an area law for Wilson loops, provided that unevaluated terms to not cancel the effect found.