Correlation Inequalities for Spin Glass in one Dimension

TL;DR

This paper establishes new correlation inequalities for the one-dimensional Edwards-Anderson spin glass model, revealing sign changes in correlations depending on disorder symmetry and parameter regions.

Contribution

It proves two novel inequalities for correlations in the 1D spin glass model, including a case with opposite sign to the classical GKS inequality.

Findings

The second inequality has the opposite sign of the GKS inequality.

In non-symmetric disorder, the truncated correlation changes sign across a specific parameter line.

Direct correlation maintains its sign in the non-symmetric case.

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.