Metastable Hierarchy in Abstract Low-Temperature Lattice Models

TL;DR

This paper reviews the hierarchical structure of metastability in low-temperature lattice models, illustrating how stable states transition between each other in a multi-level hierarchy as temperature decreases, with examples including the Ising model.

Contribution

It introduces a hierarchical decomposition framework for metastable states in abstract lattice systems and demonstrates this structure through various dynamics in the Ising model.

Findings

Existence of a metastable hierarchy with multiple levels.

Characterization of tunneling metastable transitions as temperature tends to infinity.

Application of the hierarchy to different dynamics in the Ising model.

Abstract

In this article, we review the metastable hierarchy in low-temperature lattice models. In the first part, we state that for any abstract lattice system governed by a Hamiltonian potential and evolving according to a Metropolis-type dynamics, there exists a hierarchical decomposition of the collection of stable plateaux in the system into multiple $\mathfrak{m}$ levels, such that at each level there exist tunneling metastable transitions between the stable plateaux, which can be characterized by convergence to a simple Markov chain as the inverse temperature $\beta$ tends to infinity. In the second part, we collect several examples that realize this hierarchical structure of metastability. In order to fix the ideas, we select the Ising model as our lattice system and discuss its metastable behavior under four different types of dynamics, namely the Glauber dynamics with positive/zero external fields and the Kawasaki dynamics with few/many particles. This review article is submitted to the proceedings of the event PSPDE XII, held at the University of Trieste from September 9-13, 2024.