For 2-D lattice spin systems weak mixing implies strong mixing
For 2-D lattice spin systems weak mixing implies strong mixing
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对于二维晶格自旋系统，弱混合意味着强混合
DOI：
10.1007/bf02099735
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发表时间：
1994
影响因子：
2.4
通讯作者：
R. Schonmann
中科院分区：
文献类型：
--
作者：
F. Martinelli;E. Olivieri;R. Schonmann
我们证明了对于二维晶格Z2上的有限范围离散自旋系统，从Gibbs态的Dobrushin-Shlosman唯一性条件得到的(弱)混合条件暗示了Gibbs态的更强的混合性质，类似于Dobrushin-Shlosman完全解析性条件，但仅限于晶格中的所有正方形，或者更一般地，限于足够大的正方形的所有集合的倍数。导致证明的关键观察是，边界条件的变化既不能在整体内传播，因为弱混合条件，也不能沿着边界传播，因为它是一维的。因此，对于铁磁Ising-型系统，我们得到了几个优良性质在临界温度附近任意保持的证明；这些性质包括：存在收敛的团簇展开和对数Sobolev常数的一致有界性，以及相关的Glauber动力学在包括全晶格在内的优美子集上快速收敛到平衡。
We prove that for finite range discrete spin systems on the two dimensional latticeZ2, the (weak) mixing condition which follows, for instance, from the Dobrushin-Shlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analyticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the associated Glauber dynamics on nice subsets ofZ2, including the full lattice.