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29 March 2024
 
  » arxiv » math-ph/0202013

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Some New Results on the Kinetic Ising Model in a Pure Phase
T. Bodineau ; F. Martinelli ;
Date 8 Feb 2002
Subject Mathematical Physics; Probability MSC-class: 82B10, 82B20, 60K35 | math-ph math.MP math.PR
AbstractWe consider a general class of Glauber dynamics reversible with respect to the standard Ising model in $bZ^d$ 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’’ $hat _c$ 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 $dge 2$ are then obtained via a careful use of the recent $bL^1$--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 $d=2$.
Source arXiv, math-ph/0202013
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