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 with a 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 a 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 state to the set of the stable states as β → ∞ in probability, in expectation, and in distribution. We also identify the exponent of the mixing time and find an upper bound and a lower bound for the spectral gap. We identify the minimal gates for the transition from the metastable state to the set of the stable states and for the transition from the metastable state to a fixed stable state. Furthermore, we identify the tube of typical trajectories for these two transitions. The detailed description of the energy landscape that we develop allows us to give precise asymptotics for the expected transition time from the unique metastable state to the set of the stable configurations.