Asymptotic exponential law for the transition time to equilibrium of the metastable kinetic Ising model with vanishing magnetic field

TL;DR

This paper analyzes the metastable transition times in a 2D Ising model with vanishing magnetic field, showing that the system rapidly relaxes to a metastable state before transitioning exponentially to equilibrium, with detailed capacity estimates.

Contribution

It provides a detailed asymptotic analysis of the transition time to equilibrium in the metastable Ising model under vanishing magnetic field, including pathwise descriptions and capacity estimates.

Findings

Rapid relaxation to metastable equilibrium before transition

Asymptotic exponential distribution of transition times

Capacity bounds for metastable and stable states

Abstract

We consider a Glauber dynamics associated with the Ising model on a large two-dimensional box with with minus boundary conditions and in the limit of a vanishing positive external magnetic field. The volume of this box increases quadratically in the inverse of the magnetic field. We show that at subcritical temperature and for a large class of starting measures, including measures that are supported by configurations with macroscopic plus-spin droplets, the system rapidly relaxes to some metastable equilibrium ---with typical configurations made of microscopic plus-phase droplets in a sea of minus spins--- before making a transition at an asymptotically exponential random time towards equilibrium ---with typical configurations made of microscopic minus-phase droplets in a sea of plus spins inside a large contour that separates this plus phase from the boundary. We get this result by bounding from above the local relaxation times towards metastable and stable equilibria. This makes possible to give a pathwise description of such a transition, to control the asymptotic behaviour of the mixing time in terms of soft capacities and to give estimates of these capacities.