The free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied in . In  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  and the synchronization mechanism used in the vector spin models in  and . 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.
"Free energy of multiple systems of spherical spin glasses with constrained overlaps." Electron. J. Probab. 25 1 - 34, 2020. https://doi.org/10.1214/20-EJP431