The Replica Symmetric Formula for the SK Model Revisited

TL;DR

This paper extends Bolthausen's proof of the replica symmetric formula for the SK model, establishing replica symmetry under specific conditions on parameters using a conditional second moment method.

Contribution

It provides a simplified extension of Bolthausen's proof and proves replica symmetry for all parameters satisfying a certain inequality.

Findings

Proves replica symmetry for all (eta,h) satisfying a specific inequality.

Introduces a conditional second moment method to analyze the partition function.

Simplifies the proof of the replica symmetric formula for the SK model.

Abstract

We provide a simple extension of Bolthausen's Morita type proof of the replica symmetric formula [E. Bolthausen, Stat. Mech. of Classical and Disordered Systems, pp. 63-93 (2018)] for the Sherrington-Kirkpatrick model and prove the replica symmetry for all $(\beta,h)$ that satisfy $\beta^2 E \operatorname{sech}^2(\beta\sqrt{q}Z+h) \leq 1$, where $q = E\tanh^2(\beta\sqrt{q}Z+h)$. Compared to [E. Bolthausen, Stat. Mech. of Classical and Disordered Systems, pp. 63-93 (2018)], the key of the argument is to apply the conditional second moment method to a suitably reduced partition function.