Correlation Inequalities for Spin Glass in one Dimension

Abstract

We prove two inequalities for the direct and truncated correlation for the nearest-neighboor one-dimensional Edwards-Anderson model with symmetric quenched disorder. The second inequality has the opposite sign of the GKS inequality of type II. In the non symmetric case with positive average we show that while the direct correlation keeps its sign the truncated one changes sign when crossing a suitable line in the parameter space. That line separates the regions satisfying the GKS second inequality and the one proved here.