  We present a detailed study of the interplay between chiral symmetry and
spectral properties of the Dirac operator in lattice gauge theories. We
consider, in the framework of the Schwinger model, the fixed point action and a
fermion action recently proposed by Neuberger. Both actions show the remnant of
chiral symmetry on the lattice as formulated in the Ginsparg-Wilson relation.
We check this issue for practical implementations, also evaluating the fermion
condensate in a finite volume by a subtraction procedure. Moreover, we
investigate the distribution of the eigenvalues of a properly defined
anti-hermitian lattice Dirac operator, studying the statistical properties at
the low lying edge of the spectrum. The comparison with the predictions of
chiral Random Matrix Theory enables us to obtain an estimate of the infinite
volume fermion condensate.
