Exploring Metastability in Ising models: critical droplets, energy barriers and exit time

TL;DR

This work surveys rigorous results on metastability in Ising models under Glauber dynamics across lattices, dimensions, and extensions, focusing on energy barriers, critical droplets, and exit times from metastable to stable states. It organizes methods into pathwise large deviations and potential theory, detailing how barriers Γ and gates determine transition times and typical trajectories, with sharp prefactors provided in several cases. The paper presents concrete results for 2D square and hexagonal lattices, extends to higher dimensions, anisotropic and long-range variants, and covers Kawasaki dynamics, Blume-Capel, and Potts models, illustrating how geometry and dynamics shape metastable behavior. Overall, it connects energy landscapes, geometric isoperimetry (via Peierls contours and nonlocal perimeters), and stochastic dynamics to deliver quantitative predictions for nucleation and relaxation in a broad class of Ising-type systems.

Abstract

This paper provides an overview of the research on the metastable behavior of the Ising model. We analyze the transition times from the set of metastable states to the set of the stable states by identifying the critical configurations that the system crosses with high probability during this transition and by computing the energy barrier that the system must overcome to reach the stable state starting from the metastable one. We describe the dynamical phase transition of the Ising model evolving under Glauber dynamics across various contexts, including different lattices, dimensions and anisotropic variants. The analysis is extended to related models, such as long-range Ising model, Blume-Capel and Potts models, as well as to dynamics like Kawasaki dynamics, providing insights into metastability across different systems.

Figures (17)

• Figure 1: On the left, an example of the communication height between two configurations $\sigma$ and $\eta$. On the right, the stability level of a configuration $\sigma$.
• Figure 2: In the figure, we can see the optimal path $(x \to y)_{opt}$ and the set of saddles $\mathcal{S}(x,y)=\{\omega_1,...,\omega_5\}$. In this case, the minimal gates are $\{\omega_1,\omega_2,\omega_4\}$ and $\{\omega_1,\omega_2,\omega_5\}$.
• Figure 3: The configuration $\tilde{\xi}$ is an unessential saddle, while $\xi$ is an essential saddle. Indeed, if the system follows the blue path, starting from $\sigma$ and before reaching $\eta$, it returns to $w$ and crosses $\xi$ following the black path. This figure is taken on jacquier2022metastability.
• Figure 4: The solid lines show the lattice $\mathbb{L}$, whereas the dashed lines show its dual, $\mathbb{D}$. The solid square on the left and the solid triangle on the right highlight a face of $\mathbb{D}$ centered at site $i \in \mathbb{L}$. The thicker vertices are the nearest neighbors of site $i$ on the $\mathbb{L}$.
• Figure 5: From the central panel to the left, the rectangle shrinks. From the central panel to the right, the rectangle grows.

Theorems & Definitions (5)

• Theorem 2.1: Recurrence property
• Theorem 2.2: Transition time
• Theorem 2.3: Expected value for the transition time
• Theorem 2.4: Sharp estimate for the transition time
• Theorem 2.5: Mixing time and spectral gap for Metropolis Markov chains