A new correlation inequality for Ising models with external fields

We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. show that influence boundary conditions any given spin maximised when field identically 0. One corollary spin–spin correlations are vanishes and condition free, which proves conjecture Shlosman. In particular, random model $${\mathbb {Z}}^d$$ , $$d\geqslant 3$$ exhibits exponential decay in entire high temperature regime pure model. Another satisfies conjectured strong spatial mixing property regime.