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

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).