  The standard phase-ordering process is obtained by quenching a system, like
the Ising model, to below the critical point. This is usually done with
periodic boundary conditions to insure ergodicity breaking in the low
temperature phase. With this arrangement the infinite system is known to remain
permanently out of equilibrium, i.e. there exists a well defined asymptotic
state which is time-invariant but different from the ordered ferromagnetic
state. In this paper we establish the critical nature of this invariant state,
by demonstrating numerically that the quench dynamics with periodic and
antiperiodic boundary conditions are indistinguishable one from the other.
However while the asymptotic state does not coincide with the equilibrium state
for the periodic case, it coincides instead with the equilibrium state of the
antiperiodic case, which in fact is critical. The specific example of the Ising
model is shown to be one instance of a more general phenomenon, since an
analogous picture emerges in the spherical model, where boundary conditions are
kept fixed to periodic, while the breaking or preserving of ergodicity is
managed by imposing the spherical constraint either sharply or smoothly.
