Glauber Dynamics for the mean-field Potts Model

Abstract

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 $β_s(q)$ 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 $C(β, q) n \log n$ 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 $β\uparrow β_s$ 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.

45 pages, 5 figures