Some new results on the kinetic Ising model in a pure phase

We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in ℤ^d with zero external field and inverse temperature β strictly larger than the critical value β _c in dimension 2 or the so called "slab threshold" β̂_c in dimension d ≥ 3. We first prove that the inverse spectral gap in a large cube of side N with plus boundary conditions is, apart from logarithmic corrections, larger than N in d = 2 while the logarithmic Sobolev constant is instead larger than N² in any dimension. Such a result substantially improves over all the previous existing bounds and agrees with a similar computations obtained in the framework of a one dimensional toy model based on mean curvature motion. The proof, based on a suggestion made by H. T. Yau some years ago, explicitly constructs a subtle test function which forces a large droplet of the minus phase inside the plus phase. The relevant bounds for general d ≥ 2 are then obtained via a careful use of the recent double-struck L sign¹-approach to the Wulff construction. Finally we prove that in d = 2 the probability that two independent initial configurations, distributed according to the infinite volume plus phase and evolving under any coupling, agree at the origin at time t is bounded from below by a stretched exponential exp(-√t), again apart from logarithmic corrections. Such a result should be considered as a first step toward a rigorous proof that, as conjectured by Fisher and Huse some years ago, the equilibrium time auto-correlation of the spin at the origin decays as a stretched exponential in d = 2.