  We consider the Glauber dynamics for the 2D Ising model in a box of side L,
at inverse temperature $\beta$ and random boundary conditions $\tau$ whose
distribution P either stochastically dominates the extremal plus phase (hence
the quotation marks in the title) or is stochastically dominated by the
extremal minus phase. A particular case is when P is concentrated on the
homogeneous configuration identically equal to + (equal to -). For $\beta$
large enough we show that for any $\epsilon$ there exists $c=c(\beta,\epsilon)$
such that the corresponding mixing time $T_{mix}$ satisfies
$\lim_{L\to\infty}P(T_{mix}> \exp({cL^\epsilon})) =0$. In the non-random case
$\tau\equiv +$ (or $\tau\equiv -$), this implies that $T_{mix}<
\exp({cL^\epsilon})$. The same bound holds when the boundary conditions are all
+ on three sides and all - on the remaining one. The result, although still
very far from the expected Lifshitz behaviour $T_{mix}=O(L^2)$, considerably
improves upon the previous known estimates of the form $T_{mix}\le \exp({c
L^{1/2 + \epsilon}})$. The techniques are based on induction over length
scales, combined with a judicious use of the so-called "censoring inequality"
of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to
its equilibrium measure.

Probability