Some examples of quenched self-averaging in models with Gaussian disorder

TL;DR

This paper presents an elementary approach to proving quenched disorder chaos and self-averaging phenomena in Gaussian disordered spin systems, extending previous results to various models and temperature regimes.

Contribution

It introduces a simplified method to establish quenched disorder chaos and self-averaging in multiple Gaussian disordered models, broadening the scope of prior work.

Findings

Proved quenched disorder chaos for bond overlap in Edwards-Anderson models.

Established quenched self-averaging of bond magnetization and site overlap.

Demonstrated self-averaging properties of modified random fields.

Abstract

In this paper we give an elementary approach to several results of Chatterjee in arXiv:0907.3381 and arXiv:1404.7178, as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards-Anderson type models with Gaussian disorder. The proof extends to systems at different temperatures and covers a number of other models, such as the mixed $p$-spin model, Sherrington-Kirkpatrick model with multi-dimensional spins and diluted $p$-spin model. Next, we adapt the same idea to prove quenched self-averaging of the bond magnetization for one system and use it to show quenched self-averaging of the site overlap for random field models with positively correlated spins. Finally, we show self-averaging for certain modifications of the random field itself.