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We prove Gibbs distribution of two-state spin systems(also known as binaryMarkov random fields) without hard constrains on a tree exhibits strong spatialmixing(also known as strong correlation decay), under the assumption that, forarbitrary `external field', the absolute value of `inverse temperature' issmall, or the `external field' is uniformly large or small. The first conditionon `inverse temperature' is tight if the distribution is restricted toferromagnetic or antiferromagnetic Ising models.  Thanks to Weitz's self-avoiding tree, we extends the result for sparse onaverage graphs, which generalizes part of the recent work of Mossel andSly\cite{MS08}, who proved the strong spatial mixing property for ferromagneticIsing model. Our proof yields a different approach, carefully exploiting themonotonicity of local recursion. To our best knowledge, the second condition of`external field' for strong spatial mixing in this paper is first consideredand stated in term of `maximum average degree' and `interaction energy'. As anapplication, we present an FPTAS for partition functions of two-state spinmodels without hard constrains under the above assumptions in a general familyof graphs including interesting bounded degree graphs.

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