Loop inequalities and confinement

We consider correlation inequalities that follow from the well-known loop equations of LGT, and their analogues in spin systems. They provide a way of bounding long range by short or intermediate range correlations. In several cases the method easily reproduces results that otherwise require considerable effort to obtain. In particular, in the case of the 2-dimensional O(N) spin model, where large N analytical results are available, the absence of a phase transition and the exponential decay of correlations for all $\beta$ is easily demonstrated. We report on the possible application of this technique to the analogous 4-dimensional problem of area law for the Wilson loop in LGT at large $\beta$.