Free Energy of Multiple Systems of Spherical Spin Glasses with Constrained Overlaps

TL;DR

This paper establishes the exact free energy of multiple spherical spin glass systems with constrained overlaps by proving the sharpness of the upper bound through a novel combination of the Aizenman-Sims-Starr scheme and synchronization techniques.

Contribution

It introduces a vector version of the Aizenman-Sims-Starr scheme and applies synchronization properties to precisely determine the free energy in constrained spherical spin glasses.

Findings

Proved the upper bound of the constrained free energy is sharp.

Developed a vector version of the Aizenman-Sims-Starr scheme.

Used synchronization mechanisms to match the lower bound.

Abstract

The free energy of multiple systems of spherical spin glasses with constrained overlaps was first studied in arXiv:math/0604082. The authors proved an upper bound of the constrained free energy using Guerra's interpolation. In this paper, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman-Sims-Starr scheme in arXiv:1204.5115 and the synchronization mechanism used in the vector spin models in arXiv:1512.04441 and arXiv:1512.00370 to prove the matching lower bound. We derive a vector version of the Aizenman-Sims-Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda-Guerra identities to prove the matching lower bound.