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

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

Research output: Contribution to journalArticle

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 TMIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature and any " > 0 there exists c D c.; "/ such that limL!1P.TMIX exp.cL"// D 0. In particular, for the all-plus boundary conditions and large enough, TMIX exp.cL"/. Here we show that the same conclusions hold for all larger than the critical value c and with exp.cL"/ replaced by Lc logL (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].

Original languageEnglish (US)
Pages (from-to)339-386
Number of pages48
JournalJournal of the European Mathematical Society
Volume15
Issue number2
DOIs
StatePublished - 2013

Fingerprint

Stochastic Ising Model
Mixing Time
Ising model
Criticality
Ising Model
Polynomials
Boundary conditions
Brownian Bridge
Glauber Dynamics
Polynomial
Critical value
Duality
Complement
Configuration
Line
Estimate
Temperature

Keywords

  • Glauber dynamics
  • Ising model
  • Mixing time
  • Phase coexistence

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality. / Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio.

In: Journal of the European Mathematical Society, Vol. 15, No. 2, 2013, p. 339-386.

Research output: Contribution to journalArticle

Lubetzky, Eyal ; Martinelli, Fabio ; Sly, Allan ; Toninelli, Fabio Lucio. / Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality. In: Journal of the European Mathematical Society. 2013 ; Vol. 15, No. 2. pp. 339-386.
@article{1d4cfeec6f824ab0a61c4212613c7c33,
title = "Quasi-polynomial mixing of the 2D stochastic Ising model with {"}plus{"} boundary up to criticality",
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 TMIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature and any {"} > 0 there exists c D c.; {"}/ such that limL!1P.TMIX exp.cL{"}// D 0. In particular, for the all-plus boundary conditions and large enough, TMIX exp.cL{"}/. Here we show that the same conclusions hold for all larger than the critical value c and with exp.cL{"}/ replaced by Lc logL (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].",
keywords = "Glauber dynamics, Ising model, Mixing time, Phase coexistence",
author = "Eyal Lubetzky and Fabio Martinelli and Allan Sly and Toninelli, {Fabio Lucio}",
year = "2013",
doi = "10.4171/JEMS/363",
language = "English (US)",
volume = "15",
pages = "339--386",
journal = "Journal of the European Mathematical Society",
issn = "1435-9855",
publisher = "European Mathematical Society Publishing House",
number = "2",

}

TY - JOUR

T1 - Quasi-polynomial mixing of the 2D stochastic Ising model with "plus" boundary up to criticality

AU - Lubetzky, Eyal

AU - Martinelli, Fabio

AU - Sly, Allan

AU - Toninelli, Fabio Lucio

PY - 2013

Y1 - 2013

N2 - 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 TMIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature and any " > 0 there exists c D c.; "/ such that limL!1P.TMIX exp.cL"// D 0. In particular, for the all-plus boundary conditions and large enough, TMIX exp.cL"/. Here we show that the same conclusions hold for all larger than the critical value c and with exp.cL"/ replaced by Lc logL (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].

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 TMIX is conjectured to be polynomial in L. In [37] it was shown that for a large enough inverse temperature and any " > 0 there exists c D c.; "/ such that limL!1P.TMIX exp.cL"// D 0. In particular, for the all-plus boundary conditions and large enough, TMIX exp.cL"/. Here we show that the same conclusions hold for all larger than the critical value c and with exp.cL"/ replaced by Lc logL (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].

KW - Glauber dynamics

KW - Ising model

KW - Mixing time

KW - Phase coexistence

UR - http://www.scopus.com/inward/record.url?scp=84873864813&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84873864813&partnerID=8YFLogxK

U2 - 10.4171/JEMS/363

DO - 10.4171/JEMS/363

M3 - Article

AN - SCOPUS:84873864813

VL - 15

SP - 339

EP - 386

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 2

ER -