Numerical Study of the Fundamental Modular Region in the Minimal Landau Gauge

We study numerically the so-called fundamental modular region Lambda, a region free of Gribov copies, in the minimal Landau gauge for pure SU(2) lattice gauge theory. To this end we evaluate the influence of Gribov copies on several quantities — such as the smallest eigenvalue of the Faddeev-Popov matrix, the third and the fourth derivatives of the minimizing function, and the so-called horizon function — which are used to characterize the region Lambda. Simulations are done at four different values of the coupling: beta = 0, 0.8, 1.6, 2.7, and for volumes up to 16^4. We find that typical (thermalized and gauge-fixed) configurations, including those belonging to the region Lambda, lie very close to the Gribov horizon $\mathrm{\partial }\mathrm{\Omega }$, and are characterized, in the limit of large lattice volume, by a negative-definite horizon tensor.