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Glauber dynamics for the mean-field Potts model. (English) Zbl 1254.82024

Summary: 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 \(C(\beta,q)n \log n\) 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 \(\beta \uparrow \beta _{s }\) 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.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
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