Glauber Dynamics For The Mean-Field Potts Model

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 \(\beta ↑{\beta }_s\) 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.