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Glauber Dynamics for the Mean-Field Potts Model

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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.

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Acknowledgements

This work was initiated while P.C., O.L. and A.S. were interns at the Theory Group of Microsoft Research, and they thank the Theory Group for its hospitality. P.C would like to acknowledge NSF grant CCF-1116013. The research of O.L. was supported in part by NSF grant OISE-07-30136.

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Authors and Affiliations

  1. Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544, USA

    P. Cuff

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    J. Ding

  3. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    O. Louidor

  4. Microsoft Research, One Microsoft Way, Redmond, WA, 98052-6399, USA

    E. Lubetzky & Y. Peres

  5. Department of Statistics, UC Berkeley, Berkeley, CA, 94720, USA

    A. Sly

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  1. P. Cuff
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Correspondence to O. Louidor.

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Cuff, P., Ding, J., Louidor, O. et al. Glauber Dynamics for the Mean-Field Potts Model. J Stat Phys 149, 432–477 (2012). https://doi.org/10.1007/s10955-012-0599-2

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