Finite volume Glauber dynamics in a small magnetic field

Abstract We consider Glauber dynamics on a finite cube in the multi dimensional integer lattice, which is associated with basic Ising model at a low enough temperature T=1/\beta under a positive magnetic field h. 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 in [SY97] to shed more light on the relaxation of the finite volume Glauber dynamics, when the thermodynamic parameter (\beta,h) is so near the phase transition line: { (\beta,h); \beta_c <\beta and h = 0 } that the Dobrushin-Shlosman mixing condition is no longer available.