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

TL;DR

This paper proves that the 2D stochastic Ising model with plus boundary conditions exhibits quasi-polynomial mixing times up to the critical temperature, improving previous exponential bounds and employing refined equilibrium estimates and duality techniques.

Contribution

It extends mixing time results for the 2D Ising model to all temperatures above the critical point, achieving quasi-polynomial bounds using new equilibrium estimates and contour analysis.

Findings

Mixing time is quasi-polynomial for all $eta > eta_c$

Refined bounds on Peierls contours up to criticality

Enhanced inductive scheme with equilibrium estimates

Abstract

We considerably improve upon the recent result of Martinelli and Toninelli 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 [MT] it was shown that for a large enough inverse-temperature $\beta$ and any $\epsilon >0$ there exists $c=c(\beta,\epsilon)$ such that $\lim_{L\to\infty}P(T_{mix}\geq \exp({c L^\epsilon}))=0$. In particular, for the all-plus boundary conditions and $\beta$ large enough $T_{mix} \leq \exp({c L^\epsilon})$. 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 [MT] 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 quantitatively sharpen the Brownian bridge picture established e.g. in [Greenberg-Ioffe (2005)],[Higuchi (1979)],[Hryniv (1998)].