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.