### Abstract

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L^{1} on a system of size n is O(logn). Whether in this regime there is cutoff, i. e. a sharp transition in the L^{1}-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ℤ^{2} this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (ℤ/nℤ)^{d} with periodic boundary conditions has cutoff at (d/2λ_{∞})logn, where λ_{∞} is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L^{1}-mixing to L^{2}-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e. g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.

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

Pages (from-to) | 719-755 |

Number of pages | 37 |

Journal | Inventiones Mathematicae |

Volume | 191 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*191*(3), 719-755. https://doi.org/10.1007/s00222-012-0404-5

**Cutoff for the Ising model on the lattice.** / Lubetzky, Eyal; Sly, Allan.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 191, no. 3, pp. 719-755. https://doi.org/10.1007/s00222-012-0404-5

}

TY - JOUR

T1 - Cutoff for the Ising model on the lattice

AU - Lubetzky, Eyal

AU - Sly, Allan

PY - 2013

Y1 - 2013

N2 - Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L1 on a system of size n is O(logn). Whether in this regime there is cutoff, i. e. a sharp transition in the L1-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ℤ2 this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (ℤ/nℤ)d with periodic boundary conditions has cutoff at (d/2λ∞)logn, where λ∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L1-mixing to L2-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e. g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.

AB - Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L1 on a system of size n is O(logn). Whether in this regime there is cutoff, i. e. a sharp transition in the L1-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ℤ2 this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (ℤ/nℤ)d with periodic boundary conditions has cutoff at (d/2λ∞)logn, where λ∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L1-mixing to L2-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e. g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.

UR - http://www.scopus.com/inward/record.url?scp=84874115832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874115832&partnerID=8YFLogxK

U2 - 10.1007/s00222-012-0404-5

DO - 10.1007/s00222-012-0404-5

M3 - Article

AN - SCOPUS:84874115832

VL - 191

SP - 719

EP - 755

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -