无硬约束两态自旋系统的强空间混合与近似配分函数
Strong Spatial Mixing and Approximating Partition Functions of Two-State Spin Systems without Hard Constrains
摘要：
我们证明吉布斯分布的两个状态的自旋系统（也称为二进制马尔可夫随机场）没有硬约束的树表现出强空间混合（也称为强相关衰减），假设下，对于任意的“外场”，“逆温度”的绝对值是小的，或“外场”是一致的大或小。如果分布仅限于铁磁或反铁磁伊辛模型，则关于“逆温度”的第一个条件是严格的。由于Weitz的自避免树，我们推广了平均稀疏图的结果，推广了Mossel和Sly \cite{MS08}最近的工作，他们证明了铁磁Ising模型的强空间混合性质.我们的证明产生了不同的方法，仔细利用局部递归的单调性。据我们所知，第二个条件的“外场”的强空间混合在本文中首先考虑和陈述的“最大平均度”和“相互作用能”。作为应用，在上述假设下，我们在一个包含有趣的有界度图的一般图族中，给出了无硬约束的两态自旋模型配分函数的FPTAS.
英文摘要：
We prove Gibbs distribution of two-state spin systems(also known as binary Markov random fields) without hard constrains on a tree exhibits strong spatial mixing(also known as strong correlation decay), under the assumption that, for arbitrary `external field', the absolute value of `inverse temperature' is small, or the `external field' is uniformly large or small. The first condition on `inverse temperature' is tight if the distribution is restricted to ferromagnetic or antiferromagnetic Ising models. Thanks to Weitz's self-avoiding tree, we extends the result for sparse on average graphs, which generalizes part of the recent work of Mossel and Sly\cite{MS08}, who proved the strong spatial mixing property for ferromagnetic Ising model. Our proof yields a different approach, carefully exploiting the monotonicity of local recursion. To our best knowledge, the second condition of `external field' for strong spatial mixing in this paper is first considered and stated in term of `maximum average degree' and `interaction energy'. As an application, we present an FPTAS for partition functions of two-state spin models without hard constrains under the above assumptions in a general family of graphs including interesting bounded degree graphs.
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