Finite-Volume Glauber Dynamics in a Small Magnetic Field

Abstract

We consider Glauber dynamics on a finite cube in d-dimensional lattice (d≥2), which is associated with basic Ising model at temperature T=1/β≪1 under a magnetic field h > 0. We prove that if the “effective magnetic field” is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated earlier by Schonmann and Yoshida to shed more light on the relaxation of the finite-volume Glauber dynamics when the thermodynamic parameter (β, h) is so near the phase transition line, (β, h); βc < β&h = 0, that the Dobrushin–Shlosman mixing condition is no longer available.