About the overlap distribution in mean field spin glass models

TL;DR

This paper rigorously analyzes the overlap distribution in the Sherrington-Kirkpatrick spin glass model, confirming the Parisi replica symmetry breaking picture and linking self-averaging properties to the replica symmetric Ansatz.

Contribution

It provides rigorous proofs of overlap distribution properties, confirms the Parisi picture, and extends results to short-range spin glass models.

Findings

Overlap distribution aligns with Parisi's replica symmetry breaking

Self-averaging of the order parameter implies replica symmetric Ansatz

Results extend to realistic short-range models

Abstract

We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the selfaveraging of the Edwards-Anderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the Sherrington-Kirkpatrick replica symmetric Ansatz. This extends previous results of Pastur and Shcherbina. Finally, we show how to generalize our results to realistic short range spin glass models.