有界度图上两态反铁磁自旋系统的近似算法
Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs
摘要：
在一篇开创性的论文（Weitz，2006）中，Weitz给出了一个确定性的完全多项式近似方案，用于在次数最多为d的图上计数指数加权独立集（等价地，近似统计物理学中硬核模型的配分函数），直到无限d-正则树上吉布斯测度唯一性的临界活动。最近，Sly（Sly，2010）表明，这是最优的，如果在最大度d的图上，对于大于无限d-正则树上的临界活动的活动，硬核配分函数存在FPRAS，则NP = RP。本文推广Weitz的方法，导出了任意场反铁磁Ising模型配分函数在最大d次图上的确定性全多项式逼近方案，直至d正则树上的相应临界点。我们的结果的主要成分是一个证据，两个国家的反铁磁自旋系统的d-正则树，弱空间混合意味着强空间混合。这反过来又使用了一个消息衰减参数，该参数将最近由Restrepo等人（Restrepo等人，2011）针对硬核模型提出的类似方法扩展到具有任意场的反铁磁伊辛模型的情况。通过标准对应，这些结果可以推广到任意两态反铁磁自旋系统。
英文摘要：
In a seminal paper (Weitz, 2006), Weitz gave a deterministic fully polynomial approximation scheme for count- ing exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently Sly (Sly, 2010) showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the infinite d-regular tree then NP = RP. In this paper, we extend Weitz's approach to derive a deterministic fully polynomial approx- imation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al (Restrepo et al, 2011) to the case of the anti-ferromagnetic Ising model with arbitrary field. By a standard correspondence, these results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints.
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