  We present a comparative study of the fate of an Ising ferromagnet on the
square lattice with periodic boundary conditions evolving under three different
zero-temperature dynamics. The first one is Glauber dynamics, the two other
dynamics correspond to two limits of the directed Ising model, defined by rules
that break the full symmetry of the former, yet sharing the same
Boltzmann-Gibbs distribution at stationarity. In one of these limits the
directed Ising model is reversible, in the other one it is irreversible. For
the kinetic Ising-Glauber model, several recent studies have demonstrated the
role of critical percolation to predict the probabilities for the system to
reach the ground state or to fall in a metastable state. We investigate to what
extent the predictions coming from critical percolation still apply to the two
other dynamics.
