Metastability of the three-state Potts model with general interactions

TL;DR

This paper analyzes the metastable behavior of a generalized two-dimensional three-state Potts model with complex energy landscape, focusing on transition times, mixing times, and critical configurations as temperature decreases.

Contribution

It provides a detailed characterization of metastability, transition pathways, and critical configurations for the Potts model with general interactions, extending previous models.

Findings

Metastable transition times grow exponentially with inverse temperature.

Identified all critical configurations during metastable transitions.

Established bounds on mixing times and spectral gap in low-temperature regime.

Abstract

We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants $J_{ij}>0$, where $i,j\in\{1,2,3\}$ are the possible spin values (or colors). The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. The system evolves according to a Glauber-type spin-flipping dynamics. We focus on a region of the parameter space where there are two symmetric metastable states and a stable state, and the height of a direct path between the metastable states is equal to the height of a direct path between any metastable state and the stable state. We study the metastable transition time in probability and in expectation, the mixing time of the dynamics and the spectral gap of the system when the inverse temperature $\beta$ tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition.