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We consider the ferromagnetic q-state Potts model on a finite grid graph withnon-zero external field and periodic boundary conditions. The system evolvesaccording to Glauber-type dynamics described by the Metropolis algorithm, andwe focus on the low temperature asymptotic regime. We analyze the case ofpositive external magnetic field. In this energy landscape there are $1$ stableconfiguration and $q-1$ metastable states. We study the asymptotic behavior ofthe first hitting time from any metastable state to the stable configuration as$\beta\to\infty$ in probability, in expectation, and in distribution. We alsoidentify the exponent of the mixing time and find an upper and a lower boundfor the spectral gap. We also geometrically identify the union of all minimalgates and the tube of typical trajectories for the transition from anymetastable state to the unique stable configuration.

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