Glauber Dynamics for the mean-field Potts Model

Abstract

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\ge 3$ 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.

45 pages, 5 figures

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• The mixing time evolution of Glauber dynamics for the mean-field Ising model
• Mean-field driven first-order phase transitions in systems with long-range interactions
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• Simulated Tempering and Swapping on Mean-Field Models
• Cutoff and Dynamical Phase Transition for the General Multi-component Ising Model
• Energy landscape of the two-component Curie-Weiss-Potts model with three spins
• Tunneling behavior of Ising and Potts models in the low-temperature regime
• Equi-Energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature
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• On Approximating the Potts Model with Contracting Glauber Dynamics
• Cutoff phenomenon of the Glauber dynamics for the Ising model on complete multipartite graphs in the high temperature regime
• Cutoff Phenomenon for Cyclic Dynamics on Hypercube
• Analysis of finite-size effect of infinite-range Ising model under Glauber dynamics
• A Phase Transition in Arrow's Theorem