Some New Results on the Kinetic Ising Model in a Pure Phase

Abstract

We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in \(\bbZ^d\) with zero external field and inverse temperature \(\gb\) strictly larger than the critical value \(\gb_c\) in dimension 2 or the so called ``slab threshold'' \(\hat \b_c\) in dimension \(d \geq 3\). 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 \(N^2\) 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 \(d\ge 2\) are then obtained via a careful use of the recent \(\bbL^1\)--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 \(\exp(-\sqrt{t})\), 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 \(d=2\).