Glauber Dynamics for the Mean-Field Potts Model

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 βs(q) strictly lower than the critical βc(q) for uniqueness of the thermodynamic limit. The dynamical critical βs(q) is the spinodal point marking the onset of metastability.We prove that when β<βs(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=βs(q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n4/3. For β>βs(q) the mixing time is exponentially large in n. Furthermore, as β↑βs 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 βs. 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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