Some inequalities for correlation functions of Ising models with quenched randomness

TL;DR

This paper extends correlation inequalities from ferromagnetic models to Ising spin glass models with quenched randomness, including symmetric and asymmetric interaction distributions, enhancing understanding of their correlation structures.

Contribution

It introduces new correlation inequalities for Ising models with quenched randomness, broadening previous results from local energies to pairs of spin sets and considering asymmetric distributions.

Findings

Established correlation inequalities for symmetric quenched randomness

Extended inequalities to local energies for pairs of spin sets

Derived inequalities for asymmetric interaction distributions

Abstract

Correlation inequalities have played an essential role in the analysis of ferromagnetic models but have not been established in spin glass models. In this study, we obtain some correlation inequalities for the Ising models with quenched randomness, where the distribution of the interactions is symmetric. The acquired inequalities can be regarded as an extension of the previous results, which were limited to the local energy for a spin set, to the local energy for a pair of spin sets. Besides, we also obtain some correlation inequalities for asymmetric distribution.