### Abstract

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time 1/2(1-β)^{n}log n with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n ^{3/2}). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n ^{3/2}) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _{c} = 1. In particular, we find a scaling window of order 1/√n around the critical temperature. In the high temperature regime, β = 1 - δ for some 0 < δ < 1 so that δ ^{2} n → ∞ with n, the mixing-time has order (n/δ) log(δ ^{2} n), and exhibits cutoff with constant 1/2 and window size n/δ. In the critical window, β = 1± δ, where δ ^{2} n is O(1), there is no cutoff, and the mixing-time has order n ^{3/2}. At low temperature, β = 1 + δ for δ > 0 with δ ^{2} n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order n/δ exp((3/4 + o(1))δ^{2}n).

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

Pages (from-to) | 725-764 |

Number of pages | 40 |

Journal | Communications in Mathematical Physics |

Volume | 289 |

Issue number | 2 |

DOIs | |

State | Published - Jul 2009 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Communications in Mathematical Physics*,

*289*(2), 725-764. https://doi.org/10.1007/s00220-009-0781-9

**The mixing time evolution of glauber dynamics for the mean-field ising model.** / Ding, Jian; Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 289, no. 2, pp. 725-764. https://doi.org/10.1007/s00220-009-0781-9

}

TY - JOUR

T1 - The mixing time evolution of glauber dynamics for the mean-field ising model

AU - Ding, Jian

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2009/7

Y1 - 2009/7

N2 - We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time 1/2(1-β)nlog n with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n 3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n 3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β c = 1. In particular, we find a scaling window of order 1/√n around the critical temperature. In the high temperature regime, β = 1 - δ for some 0 < δ < 1 so that δ 2 n → ∞ with n, the mixing-time has order (n/δ) log(δ 2 n), and exhibits cutoff with constant 1/2 and window size n/δ. In the critical window, β = 1± δ, where δ 2 n is O(1), there is no cutoff, and the mixing-time has order n 3/2. At low temperature, β = 1 + δ for δ > 0 with δ 2 n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order n/δ exp((3/4 + o(1))δ2n).

AB - We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time 1/2(1-β)nlog n with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n 3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n 3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β c = 1. In particular, we find a scaling window of order 1/√n around the critical temperature. In the high temperature regime, β = 1 - δ for some 0 < δ < 1 so that δ 2 n → ∞ with n, the mixing-time has order (n/δ) log(δ 2 n), and exhibits cutoff with constant 1/2 and window size n/δ. In the critical window, β = 1± δ, where δ 2 n is O(1), there is no cutoff, and the mixing-time has order n 3/2. At low temperature, β = 1 + δ for δ > 0 with δ 2 n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order n/δ exp((3/4 + o(1))δ2n).

UR - http://www.scopus.com/inward/record.url?scp=67349276257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349276257&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0781-9

DO - 10.1007/s00220-009-0781-9

M3 - Article

VL - 289

SP - 725

EP - 764

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -