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

Eyal Lubetzky; Fabio Martinelli; Allan Sly; Fabio Lucio Toninelli

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 2, page 339-386
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topLubetzky, Eyal, et al. "Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality." Journal of the European Mathematical Society 015.2 (2013): 339-386. <http://eudml.org/doc/277802>.

@article{Lubetzky2013,

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 $P$ 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 [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$L_\{\rightarrow \infty \} P(T_\{MIX\}\ge \texttt \{exp\}(cL^\epsilon ))=0$. In particular, for the all-plus boundary conditions and $\beta $ large enough $T_MIX\le cL^\epsilon $. Here we show that the same conclusions hold for all $\beta $ larger than the critical value $\beta _c$ and with exp$(cL^\epsilon )$ replaced by $L^\{c\texttt \{log\} L\}$ (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].},

author = {Lubetzky, Eyal, Martinelli, Fabio, Sly, Allan, Toninelli, Fabio Lucio},

journal = {Journal of the European Mathematical Society},

keywords = {Ising model; mixing time; phase coexistence; Glauber dynamics; Ising model; mixing time; phase co-existence; Glauber dynamics},

language = {eng},

number = {2},

pages = {339-386},

publisher = {European Mathematical Society Publishing House},

title = {Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality},

url = {http://eudml.org/doc/277802},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Lubetzky, Eyal

AU - Martinelli, Fabio

AU - Sly, Allan

AU - Toninelli, Fabio Lucio

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

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 2

SP - 339

EP - 386

AB - 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 $T_{MIX}$ 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$L_{\rightarrow \infty } P(T_{MIX}\ge \texttt {exp}(cL^\epsilon ))=0$. In particular, for the all-plus boundary conditions and $\beta $ large enough $T_MIX\le cL^\epsilon $. Here we show that the same conclusions hold for all $\beta $ larger than the critical value $\beta _c$ and with exp$(cL^\epsilon )$ replaced by $L^{c\texttt {log} L}$ (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].

LA - eng

KW - Ising model; mixing time; phase coexistence; Glauber dynamics; Ising model; mixing time; phase co-existence; Glauber dynamics

UR - http://eudml.org/doc/277802

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.