Metastability of Ising and Potts Models Without External Fields in Large Volumes at Low Temperatures

In this article, we investigate the energy landscape and metastable behavior of Ising and Potts models on two-dimensional square or hexagonal lattices in the low-temperature regime, especially in the absence of an external magnetic field. The energy landscape of these models without an external field is known to have a very large and complex saddle structure between the ground states. In the small-volume regime where the lattice is finite and fixed, the aforementioned complicated saddle structure has been successfully analyzed in Kim and Seo (Metastability of stochastic Ising and Potts models on lattices without external fields. arXiv: 2102.05565 , 2021) for two- or three-dimensional square lattices when the inverse temperature tends to infinity. In this article, we consider the large-volume regime where the size of the lattice grows to infinity. First, we establish an asymptotically sharp threshold such that the ground states are metastable if and only if the inverse temperature exceeds a threshold in a suitable sense. Then, we carry out a detailed analysis of the energy landscape and rigorously establish the Eyring–Kramers formula when the inverse temperature is sufficiently greater than the aforementioned sharp threshold. The proof relies on a detailed characterization of dead-ends appearing in the vicinity of optimal transitions between the ground states and on a combinatorial estimation of the number of configurations lying on a certain energy level.