Glauber Dynamics for the Mean-Field Potts Model

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

Research output: Contribution to journalArticle

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 2012

Fingerprint

Glauber Dynamics
Mixing Time
Mean-field Model
Potts Model
Metastability
First-order Phase Transition
Thermodynamic Limit
Critical Behavior
Mean Field
Power Law
Strictly
Uniqueness
Interpolate
Scaling
uniqueness
metastable state
marking
scaling
thermodynamics
temperature

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

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

In: Journal of Statistical Physics, Vol. 149, No. 3, 11.2012, p. 432-477.

Research output: Contribution to journalArticle

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, vol. 149, no. 3, pp. 432-477. https://doi.org/10.1007/s10955-012-0599-2
Cuff, P. ; Ding, J. ; Louidor, O. ; Lubetzky, Eyal ; Peres, Y. ; Sly, A. / Glauber Dynamics for the Mean-Field Potts Model. In: Journal of Statistical Physics. 2012 ; Vol. 149, No. 3. pp. 432-477.
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