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

TL;DR

This paper analyzes the metastable behavior and energy landscape of Ising and Potts models on large lattices at low temperatures, establishing thresholds and formulas for transition behaviors without external magnetic fields.

Contribution

It extends the understanding of metastability in large-volume Ising and Potts models, providing sharp thresholds and rigorous energy landscape analysis in the absence of external fields.

Findings

Identifies a sharp inverse temperature threshold for metastability in large lattices.

Derives the Eyring-Kramers formula for transition times above the threshold.

Provides detailed combinatorial estimates of configurations at critical energy levels.

Abstract

In this article, we investigate the energy landscape and metastable behavior of the 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 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. We first establish an asymptotically sharp threshold such that the ground states are metastable if and only if the inverse temperature is larger than the 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 larger than the previously mentioned sharp threshold. The proof relies on detailed characterization of dead-ends appearing in the vicinity of optimal transitions between ground states and on combinatorial estimation of the number of configurations lying on a certain energy level.