Metastability for the Ising model on the hexagonal lattice

TL;DR

This paper analyzes the metastable behavior of the Ising model on a hexagonal lattice under Metropolis dynamics at low temperatures, focusing on transition times, critical configurations, and geometric properties.

Contribution

It provides a detailed asymptotic analysis of transition times and a geometric description of critical configurations depending on thermodynamical parameters.

Findings

Asymptotic transition time from metastable to stable state determined.

Geometrical description of critical configurations varies with parameters.

Results on polyiamonds with maximal area and minimal perimeter.

Abstract

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the mixing time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.