### 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) |
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Pages (from-to) | 432-477 |

Number of pages | 46 |

Journal | Journal of Statistical Physics |

Volume | 149 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 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