Correlation inequalities and the decay of correlations in ferromagnets

We prove a variety of new correlation inequalities which bound intermediate distance correlations from below by long distance correlations. Typical is the following which holds for spin 1/2 nearest neighbor Ising ferromagnets: (Ssub(α)Ssub(γ)) <= Σ(deltaelement ofsub(B)) (Ssub(α)Ssub(delta))(Ssub(delta)Ssub(γ)) where B is any subset of the lattice whose removal divides the lattice into pieces with α, γ in distinct components. We describe various applications, e.g. the above inequality implies the critical exponent inequality eta 1. (orig.)