Metastable hierarchy in abstract low-temperature lattice models: an application to Kawasaki dynamics for Ising lattice gas with macroscopic number of particles

TL;DR

This paper develops a hierarchical framework for understanding metastability in low-temperature lattice models and applies it to Kawasaki dynamics in an Ising lattice gas, revealing new insights into ground state configurations.

Contribution

It introduces a general hierarchical decomposition of metastable states and characterizes the 3-level metastable hierarchy in Kawasaki dynamics for the Ising lattice gas.

Findings

Ground states form one-dimensional particle strips

Identifies tunneling transitions between ground states

Particles are more likely to form strips than squares

Abstract

This article is divided into two parts. In the first part, we study the hierarchical phenomenon of metastability in low-temperature lattice models in the most general setting. Given an abstract dynamical system governed by a Hamiltonian function, we prove that 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 an explicit simple Markov chain as the inverse temperature $\beta$ tends to infinity. In the second part, as an application, we characterize the $3$-level metastable hierarchy in Kawasaki dynamics for Ising lattice gas with macroscopic number of particles. We prove that the ground states in this model are those in which the particles line up and form a one-dimensional strip, and identify the full structure relevant to the tunneling transitions between these ground states. In particular, the results differ from the previous work [5] in that the particles in the ground states are likely to form a strip rather than a square droplet. The main tool is the resolvent approach to metastability, recently developed in [24]. Along with the analysis, we present a theorem on the sharp asymptotics of the exit distribution from cycles, which to the author's knowledge is not known in the community and therefore may be of independent interest.