A note on Talagrand's positivity principle

TL;DR

This paper generalizes Talagrand's positivity principle, showing it applies broadly beyond the SK model, including to the Aizenman-Sims-Starr interpolation, with improved conditions and smaller perturbations needed.

Contribution

It abstracts Talagrand's positivity principle from the SK model, extending its applicability and improving the conditions for ensuring positive overlaps.

Findings

Immediate application to Aizenman-Sims-Starr interpolation

Improved conditions for Ghirlanda-Guerra identities

Smaller perturbations suffice for positivity

Abstract

Talagrand's positivity principle states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs' measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifications in Talagrand's argument. In this version, for example, positivity principle immediately applies to the setting of Aizenman-Sims-Starr interpolation. Also, abstracting from the SK model improves the conditions in the Ghirlanda-Guerra identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap.