Positive-Overlap Transition and Critical Exponents in Mean Field Spin Glasses

TL;DR

This paper investigates the behavior of overlaps in the SK spin glass model, proving positivity under certain conditions and determining critical exponents of overlap correlations at the phase transition.

Contribution

It introduces a unified approach to show positive overlap averages and derive critical exponents without relying on Ghirlanda-Guerra identities.

Findings

Overlap averages become positive after external field application and removal.

Critical exponents of all overlap correlation functions are determined.

Overlap behavior at the critical point is rigorously characterized.

Abstract

In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is non zero. More specificly, the average of the overlap, which is naively expected to take values in the whole interval $[-1,+1]$, becomes positive if we ``first'' apply an external field, so to destroy the gauge invariance of the model, and ``then'' remove it in the thermodynamic limit. This phenomenon emerges at the critical point. This first result is weaker that the one obtained by Talagrand (not limited to the average of the overlap), but we show here that, at least in average, the overlap is proven to be non-negative with no use of the Ghirlanda-Guerra identities. The latter are instead needed to obtain the second result, which is the control the behavior of the overlap at the critical point: we find the critical exponents of all the overlap correlation functions.