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 Weitzs self-avoiding tree, we extends the result for sparse on average graphs, which generalizes part of the recent work of Mossel and Slycite{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.
Download