On the Mixed Even-Spin Sherrington-Kirkpatrick Model with Ferromagnetic Interaction

TL;DR

This paper investigates a spin system combining mixed even-spin SK couplings with Curie-Weiss interaction, deriving a variational formula for free energy, analyzing overlap positivity, and establishing magnetization bounds under various conditions.

Contribution

It introduces a variational formula for the free energy of the combined model and analyzes overlap positivity and magnetization behavior with external fields.

Findings

Thermodynamic limit of free energy given by a variational formula.

Positivity of overlap and Ghirlanda-Guerra identities hold in certain conditions.

Constructed temperature regions with controlled magnetization.

Abstract

We study a spin system with both mixed even-spin Sherrington-Kirkpatrick (SK) couplings and Curie-Weiss (CW) interaction. Our main results are: (i) The thermodynamic limit of the free energy is given by a variational formula involving the free energy of the SK model with a change in the external field. (ii) In the presence of a centered Gaussian external field, the positivity of the overlap and the extended Ghirlanda-Guerra identities hold on a dense subset of the temperature parameters. (iii) We establish a general inequality between the magnetization and overlap. (iv) We construct a temperature region in which the magnetization can be quantitatively controlled and deduce different senses of convergence for the magnetization depending on whether the external field is present or not. Our approach is based on techniques from the study of the CW and SK models and results in convex analysis.