Replica symmetry up to the de Almeida-Thouless line in the Sherrington-Kirkpatrick model

TL;DR

This paper proves that replica symmetry persists in the Sherrington-Kirkpatrick model under certain conditions, confirming a long-standing prediction by de Almeida and Thouless.

Contribution

It provides a rigorous proof of replica symmetry in the SK model up to the de Almeida-Thouless line using Parisi measure analysis.

Findings

Replica symmetry holds when eta^2 E[sech^4(eta(eta\u001a Z+h)]  1.

Confirms the de Almeida-Thouless prediction from 1978.

Uses a direct analysis of the Parisi measure for the proof.

Abstract

We show that in the Sherrington-Kirkpatrick model at inverse temperature $\beta$ with uniform external field $h>0$, replica symmetry holds in the regime $ \beta^2\mathrm{E}[ \mathrm{sech}^4(\beta\sqrt{q}Z+h)] \le 1$, where $Z$ is a standard Gaussian random variable. This confirms a prediction of de Almeida and Thouless (1978). The proof proceeds by a direct analysis of the Parisi measure using the characterization provided by Jagannath and Tobasco (2017).