### Abstract

We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog n), whereas at low temperature (β>1) it is exp (Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order 1/√ around the critical temperature β_{c}=1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ^{2}n→∞, then the mixing-time has order (n/δ)log (δ^{2}n). The cutoff constant is (1/2+[2(ζ^{2}β/δ-1)]^{-1}), where ζ is the unique positive root of g(x)=tanh (βx)-x, and the cutoff window has order n/δ.

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

Pages (from-to) | 407-458 |

Number of pages | 52 |

Journal | Journal of Statistical Physics |

Volume | 137 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2009 |

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### Keywords

- Censored dynamics
- Curie-Weiss model
- Glauber dynamics for Ising model
- Mixing time

### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*137*(3), 407-458. https://doi.org/10.1007/s10955-009-9859-1

**Censored glauber dynamics for the mean field ising model.** / Ding, Jian; Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 137, no. 3, pp. 407-458. https://doi.org/10.1007/s10955-009-9859-1

}

TY - JOUR

T1 - Censored glauber dynamics for the mean field ising model

AU - Ding, Jian

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2009/11

Y1 - 2009/11

N2 - We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog n), whereas at low temperature (β>1) it is exp (Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order 1/√ around the critical temperature βc=1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ2n→∞, then the mixing-time has order (n/δ)log (δ2n). The cutoff constant is (1/2+[2(ζ2β/δ-1)]-1), where ζ is the unique positive root of g(x)=tanh (βx)-x, and the cutoff window has order n/δ.

AB - We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog n), whereas at low temperature (β>1) it is exp (Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled. In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order 1/√ around the critical temperature βc=1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging. In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ2n→∞, then the mixing-time has order (n/δ)log (δ2n). The cutoff constant is (1/2+[2(ζ2β/δ-1)]-1), where ζ is the unique positive root of g(x)=tanh (βx)-x, and the cutoff window has order n/δ.

KW - Censored dynamics

KW - Curie-Weiss model

KW - Glauber dynamics for Ising model

KW - Mixing time

UR - http://www.scopus.com/inward/record.url?scp=76149115214&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76149115214&partnerID=8YFLogxK

U2 - 10.1007/s10955-009-9859-1

DO - 10.1007/s10955-009-9859-1

M3 - Article

VL - 137

SP - 407

EP - 458

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -