Energy Landscape and Metastability of Stochastic Ising and Potts Models on Three-dimensional Lattices Without External Fields

Abstract

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-deviationtype 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.

Energy Landscape and Metastability of Curie-Weis-Potts Model

“…Although the present work focuses on the Potts model on complete graphs, we also note that the Ising and Potts models on the lattice are widely studied as well. For instance, we refer to [34] and the references therein for the phase transition, to [26][27][28] for the cut-off phenomenon in the high-temperature regime, and to [1,4,[7][8][9][10]20,[29][30][31][32] for the metastability in the low-temperature regime.…”

Metastability of Ising and Potts models without external fields in large volumes at low temperatures

“…Finally, in the paper [20] by the authors of the present work, a complete characterization of the entire saddle structure has been carried out on fixed two or three dimensional square lattices 2 with periodic or open boundary conditions. This level of detailed understanding of the energy landscape enables us to deduce the Eyring-Kramers formula in the very low temperature regime.…”

“…[22] for the square lattice and Theorem 3.1 of the present work for the hexagonal lattice) that the energy barrier between ground states is 2L + 2. In the small volume regime considered in [7,20,22], we can neglect all the configurations of energy larger than 2L + 2 since the number of such configurations is determined solely by L (and hence fixed) and therefore, as β → ∞, these configurations have exponentially negligible mass with respect to the Gibbs measure, compared to the configurations with energy less than or equal to 2L + 2 along which typical metastable transitions take place. However, for the models in the large volume regime, we are no longer able to neglect these configurations, as the number of such configurations also grows to infinity and hence the entropy plays a role.…”

“…The analysis of canonical configurations is rather straightforward, but we need to fully understand the complex structure of the dead-ends in order to obtain quantitative results such as the Eyring-Kramers formula. In the two-dimensional square lattice, it is observed in [20] that the dead-ends are attached only at the edge part of the saddle plateau thanks to its special local geometry, and this feature allows us to avoid serious difficulties arising from the analysis of dead-ends. However, for other forms of general lattices, this miracle does not occur and the dead-ends are attached along the whole saddle plateau, so that they form a highly complicated maze structure.…”

“…The energy landscape of these models without an external field is known to have a huge and complex saddle structure between ground states. In the small volume regime where the lattice is finite and fixed, the aforementioned complicated saddle structure has been successfully analyzed in [20] 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.…”