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 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.