Disorder chaos in the Sherrington-Kirkpatrick model with external field

TL;DR

This paper proves that the Sherrington-Kirkpatrick model exhibits disorder chaos even with an external field, showing the overlap concentrates at a value determined by Guerra's bound and the Parisi measure.

Contribution

It extends the disorder chaos proof for the SK model to include external fields using Guerra's replica symmetry breaking bound.

Findings

Disorder chaos holds in the SK model with external field.

Overlap concentration is characterized by an equation involving Guerra's bound.

The position of the overlap is determined by the Parisi measure.

Abstract

We consider a spin system obtained by coupling two distinct Sherrington-Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of external field, this statement was first confirmed by Chatterjee [Disorder chaos and multiple valleys in spin glasses (2009) Preprint]. In the present paper, using Guerra's replica symmetry breaking bound, we prove that the SK model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to Guerra's bound and the Parisi measure.