Strong Spatial Mixing and Approximating Partition Functions of Two-State Spin Systems without Hard Constrains

Abstract: 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 ofinverse temperature' is small, or the external field' is uniformly large or small. The first condition oninverse 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 ofmaximum 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.