On weak mixing in lattice models
On weak mixing in lattice models
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DOI：
10.1007/s004400050155
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发表时间：
1998-04-01
影响因子：
2
通讯作者：
Alexander, KS
中科院分区：
文献类型：
--
作者：
Alexander, KS
对于Z（d）上的格点模型，弱混合是边界条件对有限区域的影响随离该区域的距离呈指数衰减的性质。对于Z（2）上的一大类模型，包括所有有限范围模型，我们证明了弱混合是吉布斯唯一性、适当形式的连通性的指数衰减和自然耦合性质的结果。特别地，在Z（2）上，Fortuin-Kasteleyn随机簇模型在唯一性保持且连通性指数衰减时是弱混合的，并且在临界温度以上的q状态Potts模型在相关性指数衰减时是弱混合的，如果q足够大，则满足假设。比率弱混合是指事件A和B分别在格的子集Lambda和Gamma上一致发生的性质，\P（A boolean AND B）/P（A）P（B）- 1\在Lambda和Gamma之间的距离上呈指数下降。我们证明了在温和的假设下，例如有限值域，弱混合蕴涵比弱混合。
For lattice models on Z(d), weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on Z(2), including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on Z(2), the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring On subsets Lambda and Gamma, respectively, of the lattice, \P(A boolean AND B)/P(A)P(B) - 1\ decreases exponentially in the distance between Lambda and Gamma. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing.