  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.
