### Abstract

On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature β is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β > 0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree d, the Ising model has cutoff provided that β < κ/d for some absolute constant κ (a result which, up to the value of κ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1), just as when β = 0. Finally, the mixing time from almost every initial state is not more than a factor of 1 + ∈_{β} faster then the worst one (with ∈_{β} →0 as β →0), whereas the uniform starting state is at least 2 -∈_{β} times faster.

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

Pages (from-to) | 3664-3696 |

Number of pages | 33 |

Journal | Annals of Probability |

Volume | 45 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2017 |

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### Keywords

- Cutoff phenomenon
- Ising model
- Mixing times of Markov chains

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*45*(6), 3664-3696. https://doi.org/10.1214/16-AOP1146

**Universality of cutoff for the ising model.** / Lubetzky, Eyal; Sly, Allan.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 45, no. 6, pp. 3664-3696. https://doi.org/10.1214/16-AOP1146

}

TY - JOUR

T1 - Universality of cutoff for the ising model

AU - Lubetzky, Eyal

AU - Sly, Allan

PY - 2017/11/1

Y1 - 2017/11/1

N2 - On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature β is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β > 0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree d, the Ising model has cutoff provided that β < κ/d for some absolute constant κ (a result which, up to the value of κ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1), just as when β = 0. Finally, the mixing time from almost every initial state is not more than a factor of 1 + ∈β faster then the worst one (with ∈β →0 as β →0), whereas the uniform starting state is at least 2 -∈β times faster.

AB - On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature β is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β > 0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree d, the Ising model has cutoff provided that β < κ/d for some absolute constant κ (a result which, up to the value of κ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1), just as when β = 0. Finally, the mixing time from almost every initial state is not more than a factor of 1 + ∈β faster then the worst one (with ∈β →0 as β →0), whereas the uniform starting state is at least 2 -∈β times faster.

KW - Cutoff phenomenon

KW - Ising model

KW - Mixing times of Markov chains

UR - http://www.scopus.com/inward/record.url?scp=85035086833&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035086833&partnerID=8YFLogxK

U2 - 10.1214/16-AOP1146

DO - 10.1214/16-AOP1146

M3 - Article

AN - SCOPUS:85035086833

VL - 45

SP - 3664

EP - 3696

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 6

ER -