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


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 LL at low temperature and with random boundary conditions whose distribution \bP\bP stochastically dominates the extremal plus phase. An important special case is when \bP\bP is concentrated on the homogeneous all-plus configuration, where the mixing time \tmix\tmix is conjectured to be polynomial in LL. In~\cite{cf
} it was shown that for a large enough inverse-temperature \b\b and any \gep>0\gep >0 there exists c=c(\b,\gep)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 β\beta large enough \tmixexp(cL\gep)\tmix\le \exp({cL^\gep}). Here we show that the same conclusions hold for all β\beta larger than the critical value βc\beta_c and with exp(cL\gep)\exp({cL^\gep}) replaced by LclogLL^{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

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