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

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 TMIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature and any " > 0 there exists c D c.; "/ such that limL!1P.TMIX exp.cL"// D 0. In particular, for the all-plus boundary conditions and large enough, TMIX exp.cL"/. Here we show that the same conclusions hold for all larger than the critical value c and with exp.cL"/ replaced by Lc logL (i.e. quasi-polynomial mixing). The key point is a modification of the inductive scheme of [37] together with refined equilibrium estimates that hold up to criticality, obtained via duality and random-line representation tools for the Ising model. In particular, we establish new precise bounds on the law of Peierls contours which complement the Brownian bridge picture established e.g. in [20, 22, 23].