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

Lubetzky, 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 -