  We consider the Glauber dynamics for the Ising model with "+" boundary
conditions, at zero temperature or at temperature which goes to zero with the
system size (hence the quotation marks in the title). In dimension d=3 we prove
that an initial domain of linear size L of "-" spins disappears within a time
\tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0.
The proof of the upper bound proceeds via comparison with an auxiliary dynamics
which mimics the motion by mean curvature that is expected to describe, on
large time-scales, the evolution of the interface between "+" and "-" domains.
The analysis of the auxiliary dynamics requires recent results on the
fluctuations of the height function associated to dimer coverings of the
infinite honeycomb lattice. Our result, apart from the spurious logarithmic
factor, is the first rigorous confirmation of the expected behavior
\tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension
d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections:
the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here
we provide the lower bound. For d=2, we also prove that the spectral gap of the
generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli,
2002].
