We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.

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.

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