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We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{MIX}$ is conjectured to be polynomial in $L$. In [37] it was shown that for a large enough inverse-temperature $\beta$ and any $ϵ>0$ there...

Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of ${\mathbb{Z}}^2$. The model describes a crystal surface by assigning a non-negative integer height ${\eta }_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $\eta$ is proportional to $\exp (-\beta \mathscr{H}(\eta \left)\right)$, where $\beta$ is the inverse-temperature and $\mathscr{H}\left(\eta \right)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures....