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We considerably improve upon the recent result of Martinelli and Toninelli onthe 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 iswhen $P$ is concentrated on the homogeneous all-plus configuration, where themixing time $T_{mix}$ is conjectured to be polynomial in $L$. In [MT] it wasshown that for a large enough inverse-temperature $\beta$ and any $\epsilon >0$there exists $c=c(\beta,\epsilon)$ such that $\lim_{L\to\infty}P(T_{mix}\geq\exp({c L^\epsilon}))=0$. In particular, for the all-plus boundary conditionsand $\beta$ large enough $T_{mix} \leq \exp({c L^\epsilon})$.  Here we show that the same conclusions hold for all $\beta$ larger than thecritical value $\beta_c$ and with $\exp({c L^\epsilon})$ replaced by $L^{c \logL}$ (i.e. quasi-polynomial mixing). The key point is a modification of theinductive scheme of [MT] together with refined equilibrium estimates that holdup to criticality, obtained via duality and random-line representation toolsfor the Ising model. In particular, we establish new precise bounds on the lawof Peierls contours which quantitatively sharpen the Brownian bridge pictureestablished e.g. in [Greenberg-Ioffe (2005)],[Higuchi (1979)],[Hryniv (1998)].

Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality