Abstract
We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in ℤd 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” β 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 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 ≥ 2 are then obtained via a careful use of the recent \(\mathbb{L}^{1}\)–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.
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Bodineau, T., Martinelli, F. Some New Results on the Kinetic Ising Model in a Pure Phase. Journal of Statistical Physics 109, 207–235 (2002). https://doi.org/10.1023/A:1019939712267
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DOI: https://doi.org/10.1023/A:1019939712267