纯相动力学伊辛模型的一些新结果
Some New Results on the Kinetic Ising Model in a Pure Phase
摘要：
我们考虑一类广义Glauber动力学在$\bbZ^d$中相对于标准伊辛模型是可逆的，其外磁场为零，逆温度$\gb$严格大于2维的临界值$\gb_c$或所谓的$d \geq 3$维的“板阈值”$\hat \b_c$.我们首先证明，在一个大的立方体的侧$N$加上边界条件的逆谱间隙是，除了对数校正，大于$N$在$d=2$，而对数Sobolev常数，而不是大于$N^2$在任何维度。这样的结果大大改善了所有以前现有的界限，并同意在一个框架中获得的一个类似的计算的基础上平均曲率运动的一维玩具模型。该证明基于H. T.几年前，Yau明确地构建了一个微妙的测试函数，该函数迫使负相的大液滴进入正相。一般$d\ge 2$的相关界限，然后通过仔细使用最近的$\bbL^1$ -方法的武尔夫建设。最后，我们证明，在$d=2$的概率，两个独立的初始配置，分布根据无限体积加相位和发展下的任何耦合，同意在原点的时间$t$是有界的从下面的拉伸指数$\exp(-\sqrt{t})$，再次除了对数校正。这样的结果应该被看作是严格证明的第一步，正如Fisher和Huse几年前所指出的那样，在$d=2$中，原点自旋的平衡时间自相关以拉伸指数衰减。
英文摘要：
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 $d=2$.
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