Crossing the Gribov horizon: an unconventional study of geometric properties of gauge-configuration space in Landau gauge

We prove a lower bound for the smallest nonzero eigenvalue of the Landau-gauge Faddeev-Popov matrix in Yang-Mills theories. The bound is written in terms of the smallest nonzero momentum on the lattice and of a parameter characterizing the geometry of the first Gribov region. This allows a simple and intuitive description of the infinite-volume limit in the ghost sector. In particular, we show how nonperturbative effects may be quantified by the rate at which typical thermalized and gauge-fixed configurations approach the Gribov horizon. Our analytic results are verified numerically in the SU(2) case through an informal, free and easy, approach. This analysis provides the first concrete explanation of why the so-called scaling solution of the Dyson-Schwinger equations is not observed in lattice studies.