Correlation inequalities and the decay of correlations in ferromagnets

Abstract

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:

$$\langle S_\alpha S_\gamma \rangle \leqq \sum\limits_{\delta \in B} {\langle S_\alpha S_\delta \rangle } \langle S_\delta S_\gamma \rangle$$

whereB 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 η<1.