Fundamental modular region, Boltzmann factor and area law in lattice 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 lattice Faddeev-Popov propagator, which appears in the modified action, acquires a dipole singularity at zero momentum, characteristic of long range correlation. This is sufficient to produce an area law for Wilson loops, provided that unevaluated terms do not cancel the effect found.