Metastability for the degenerate Potts Model with positive 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 positive external magnetic field associated to one spin value. In this energy landscape there is one stable configuration and q−1 metastable states. We study the asymptotic behavior of the first hitting time from any metastable state to the stable configuration as β→∞ 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. Finally, we identify all the minimal gates and the tube of typical trajectories for the transition from any metastable state to the unique stable configuration by giving a geometric characterization.