Strong Spatial Mixing for Binary Markov Random Fields

The remarkable contribution by Weitz gives a general framework to establish the strong spatial mixing property of Gibbs measures. In light of Weitz’s work, we prove the strong spatial mixing for binary Markov random fields under the condition that the ‘external field’ is uniformly large or small by turning them into a corresponding Ising model. Our proof is done through a ‘path’ characterization of the Lipchitz method and recursive formula on trees, which enables us to combine the idea of the self-avoiding tree.