  We investigate analytically and numerically a random-matrix model for m
fermions occupying l1 single-particle states with positive parity and l2
single-particle states with negative parity and interacting through random
two-body forces that conserve parity. The single-particle states are completely
degenerate and carry no further quantum numbers. We compare spectra of
many-body states with positive and with negative parity. We show that in the
dilute limit, ground states with positive and with negative parity occur with
equal probability. Differences in the ground-state probabilities are, thus, a
finite-size effect and are mainly due to different dimensions of the Hilbert
spaces of either parity.
