Glauber Dynamics for the mean-field Potts Model

TL;DR

This paper analyzes the Glauber dynamics for the mean-field Potts model with multiple states, revealing a critical slowdown at a spinodal point below the thermodynamic critical temperature, with detailed mixing time behaviors across different regimes.

Contribution

It provides the first comprehensive analysis of mixing times and cutoff phenomena around the critical dynamical temperature for a model with a first order phase transition.

Findings

Mixing time is $C(eta, q) n \,\log n$ for $eta<\beta_s(q)$.

At $eta=\beta_s(q)$, mixing follows a power-law of order $n^{4/3}$.

For $eta>\beta_s(q)$, mixing time becomes exponentially large in $n$.

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 $\beta_s(q)$ 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.