### Abstract

The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on ℤ ^{2} everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical β _{c} it is polynomial in the side-length and at low temperature it is exponential. A seminal series of papers verified this on ℤ ^{2} except at β = β _{c} where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in ℤ ^{2}. Namely, we show that on a finite box with arbitrary (e. g. fixed, free, periodic) boundary conditions, the inverse-gap at β = β _{c} is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of the critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

Original language | English (US) |
---|---|

Pages (from-to) | 815-836 |

Number of pages | 22 |

Journal | Communications in Mathematical Physics |

Volume | 313 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2012 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*313*(3), 815-836. https://doi.org/10.1007/s00220-012-1460-9

**Critical Ising on the Square Lattice Mixes in Polynomial Time.** / Lubetzky, Eyal; Sly, Allan.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 313, no. 3, pp. 815-836. https://doi.org/10.1007/s00220-012-1460-9

}

TY - JOUR

T1 - Critical Ising on the Square Lattice Mixes in Polynomial Time

AU - Lubetzky, Eyal

AU - Sly, Allan

PY - 2012/8

Y1 - 2012/8

N2 - The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on ℤ 2 everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical β c it is polynomial in the side-length and at low temperature it is exponential. A seminal series of papers verified this on ℤ 2 except at β = β c where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in ℤ 2. Namely, we show that on a finite box with arbitrary (e. g. fixed, free, periodic) boundary conditions, the inverse-gap at β = β c is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of the critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

AB - The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on ℤ 2 everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical β c it is polynomial in the side-length and at low temperature it is exponential. A seminal series of papers verified this on ℤ 2 except at β = β c where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in ℤ 2. Namely, we show that on a finite box with arbitrary (e. g. fixed, free, periodic) boundary conditions, the inverse-gap at β = β c is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of the critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

UR - http://www.scopus.com/inward/record.url?scp=84864355145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864355145&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1460-9

DO - 10.1007/s00220-012-1460-9

M3 - Article

AN - SCOPUS:84864355145

VL - 313

SP - 815

EP - 836

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -