### Abstract

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β _{s}(q) strictly lower than the critical β _{c}(q) for uniqueness of the thermodynamic limit. The dynamical critical β _{s}(q) is the spinodal point marking the onset of metastability. We prove that when β<β _{s}(q) the mixing time is asymptotically C(β,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At β=β _{s}(q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^{4/3}. For β>β _{s}(q) the mixing time is exponentially large in n. Furthermore, as β↑β _{s} with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^{-2/3}) around β _{s}. These results form the first complete analysis of mixing around the critical dynamical temperature-including the critical power law-for a model with a first order phase transition.

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

Pages (from-to) | 432-477 |

Number of pages | 46 |

Journal | Journal of Statistical Physics |

Volume | 149 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2012 |

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### Keywords

- Critical slowdown
- Curie Weiss
- Cutoff
- Glauber dynamics
- Mean field
- Metastability
- Mixing time
- Potts model
- Spinodal point

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*149*(3), 432-477. https://doi.org/10.1007/s10955-012-0599-2

**Glauber Dynamics for the Mean-Field Potts Model.** / Cuff, P.; Ding, J.; Louidor, O.; Lubetzky, Eyal; Peres, Y.; Sly, A.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 149, no. 3, pp. 432-477. https://doi.org/10.1007/s10955-012-0599-2

}

TY - JOUR

T1 - Glauber Dynamics for the Mean-Field Potts Model

AU - Cuff, P.

AU - Ding, J.

AU - Louidor, O.

AU - Lubetzky, Eyal

AU - Peres, Y.

AU - Sly, A.

PY - 2012/11

Y1 - 2012/11

N2 - We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β s(q) strictly lower than the critical β c(q) for uniqueness of the thermodynamic limit. The dynamical critical β s(q) is the spinodal point marking the onset of metastability. We prove that when β<β s(q) the mixing time is asymptotically C(β,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At β=β s(q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n 4/3. For β>β s(q) the mixing time is exponentially large in n. Furthermore, as β↑β s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n -2/3) around β s. These results form the first complete analysis of mixing around the critical dynamical temperature-including the critical power law-for a model with a first order phase transition.

AB - We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β s(q) strictly lower than the critical β c(q) for uniqueness of the thermodynamic limit. The dynamical critical β s(q) is the spinodal point marking the onset of metastability. We prove that when β<β s(q) the mixing time is asymptotically C(β,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At β=β s(q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n 4/3. For β>β s(q) the mixing time is exponentially large in n. Furthermore, as β↑β s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n -2/3) around β s. These results form the first complete analysis of mixing around the critical dynamical temperature-including the critical power law-for a model with a first order phase transition.

KW - Critical slowdown

KW - Curie Weiss

KW - Cutoff

KW - Glauber dynamics

KW - Mean field

KW - Metastability

KW - Mixing time

KW - Potts model

KW - Spinodal point

UR - http://www.scopus.com/inward/record.url?scp=84867989702&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867989702&partnerID=8YFLogxK

U2 - 10.1007/s10955-012-0599-2

DO - 10.1007/s10955-012-0599-2

M3 - Article

AN - SCOPUS:84867989702

VL - 149

SP - 432

EP - 477

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -