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

### Eyal Lubetzky

New York University, USA### Fabio Martinelli

Università degli Studi Roma Tre, Italy### Allan Sly

University of California at Berkeley, USA### Fabio L. Toninelli

École Normale Supérieure de Lyon, France

## Abstract

We considerably improve upon the recent result of~\cite{cf

} on the mixing time of Glauber dynamics for the \twod Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $\bP$ stochastically dominates the extremal plus phase. An important special case is when $\bP$ is concentrated on the homogeneous all-plus configuration, where the mixing time $\tmix$ is conjectured to be polynomial in $L$. In~\cite{cf} it was shown that for a large enough inverse-temperature $\b$ and any $\gep >0$ there exists $c=c(\b,\gep)$ such that \lim_{L\to\infty}\bP\(\tmix\ge \exp({cL^\gep})\)=0. In particular, for the all-plus boundary conditions and $\beta$ large enough $\tmix\le \exp({cL^\gep})$. Here we show that the same conclusions hold for all $\beta$ larger than the critical value $\beta_c$ and with $\exp({cL^\gep})$ replaced by $L^{c \log L }$ (\ie quasi-polynomial mixing). The key point is a modification of the inductive scheme of~\cite{cf} 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~\cites{cf,cf,cf}.## Cite this article

Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio L. Toninelli, Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality. J. Eur. Math. Soc. 15 (2013), no. 2, pp. 339–386

DOI 10.4171/JEMS/363