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

Abstract

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 $\mathbf{P}$ stochastically dominates the extremal plus phase. An important special case is when $\mathbf{P}$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{\mathrm{MIX}}$ is conjectured to be polynomial in $L$. In [37] it was shown that for a large enough inverse-temperature $\beta$ and any $\epsilon >0$ there exists $c=c(\beta ,\epsilon )$ such that ${\lim }_{L\to \infty }\mathbf{P}(T_{\mathrm{MIX}}\ge \exp (c{L}^{\epsilon }))=0$. In particular, for the all-plus boundary conditions and $\beta$ large enough $T_{\mathrm{MIX}}\le \exp (c{L}^{\epsilon })$. Here we show that the same conclusions hold for all $\beta$ larger than the critical value ${\beta }_c$ and with $\exp (c{L}^{\epsilon })$ replaced by $L^{c\log L}$ (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].