We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with \\(q\\geq 3\\) states and show that it undergoes a critical slowdown at an inverse-temperature \\({\\beta }_s(q)\\) strictly lower than the critical \\({\\beta }_c(q)\\) for uniqueness of the thermodynamic limit. The dynamical critical \\({\\beta }_s(q)\\) is the spinodal point marking the onset of metastability. We prove that when \\(\\beta <{\\beta }_s(q)\\) the mixing time is asymptotically \\(C(\\beta ,q)nlogn\\) and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order \\(n\\). At \\(\\beta ={\\beta }_s(q)\\) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order \\(n_{4\/3}\\). For \\(\\beta >{\\beta }_s(q)\\) the mixing time is exponentially large in \\(n\\). Furthermore, as \\(\\beta \u2191{\\beta }_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 \\({\\beta }_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.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with states and show that it undergoes a critical slowdown at an inverse-temperature strictly lower than the critical for uniqueness of the thermodynamic limit. The dynamical critical is the spinodal point marking the onset of metastability. 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