Glauber Dynamics for the Mean-Field Potts Model

P. Cuff, J. Ding, O. Louidor, E. Lubetzky, Y. Peres, A. Sly

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22 Scopus citations

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 languageEnglish (US)
Pages (from-to)432-477
Number of pages46
JournalJournal of Statistical Physics
Volume149
Issue number3
DOIs
StatePublished - Nov 1 2012

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

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

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  • Cite this

    Cuff, P., Ding, J., Louidor, O., Lubetzky, E., Peres, Y., & Sly, A. (2012). Glauber Dynamics for the Mean-Field Potts Model. Journal of Statistical Physics, 149(3), 432-477. https://doi.org/10.1007/s10955-012-0599-2