Metastability of Stochastic Ising and Potts Models on Lattices without External Fields.

In this article, we investigate the metastable behaviors of the Glauber dynamics associated with Ising and Potts models on fixed, finite, but large lattices of dimensions two or three in the very-low-temperature regime. Metastability analyses of these models for non-zero external magnetic fields have been extensively researched during the last two decades; however, models without external fields remained uninvestigated. Recently, [Nardi and Zocca, Stochastic Processes and thier Applications, 129: 4556-4575, 2019] analyzed for the first time the two-dimensional case and obtained several large-deviation-type results for the metastability, by implementing the so-called pathwise approach. In the current article, we comprehensively extend these results in two ways. First, we obtain more refined results, including the Eyring-Kramers law, Markov-chain-model reductions, and a full characterization of metastable transition paths for the two-dimensional model. Second, we perform the corresponding analyses for the three-dimensional case. To this end, we derive precise descriptions of the energy landscape, which is particularly complex in the three-dimensional case, and then analyze the typical behavior of the Glauber dynamics on a suitable neighborhood of a very large set of complicated saddle configurations.