Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics

Abstract

We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of negative external magnetic field. In this scenario there are $q-1$ stable configurations and a unique metastable state. We describe the asymptotic behavior of the first hitting time from the metastable to the set of the stable states as $\beta \to \infty$ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper and a lower bound for the spectral gap. We identify the union of all minimal gates for the transition from the metastable state both to the set of the stable states and to a fixed stable state. Furthermore, we identify the tube of typical trajectories for these two transitions. The accurate knowledge of the energy landscape allows us to give precise asymptotics for the expected transition time from the unique metastable state to the set of the stable configurations.

45 pages, 17 figures

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• Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata
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• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions
• Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field
• Metastability for Kawasaki dynamics on the hexagonal lattice
• Metastability of Ising and Potts models without external fields in large volumes at low temperatures
• Metastability of the three-state Potts model with general interactions
• Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics
• Critical configurations of the hard-core model on square grid graphs