Exponential relaxation of finite volume Glauber dynamics near the border of the one phase region

Abstract We consider Glauber dynamics on a finite set in the multi-dimensional integer lattice, which is associated with basic Ising model. We are particularly interested in the case that the thermodynamic parameter (\beta, h) is sitting in a place in the one phase region, which is so near to the multi phase region that the Dobrushin-Shlosman mixing condition is no longer available. We prove that for some ``nice" boundary conditions, the convergence of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying subset in the lattice.