Free energy of multiple systems of spherical spin glasses with constrained overlaps

Abstract

The free energy of multiple systems of spherical spin glasses with constrained overlaps was first studied in [22]. 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 [4] and the synchronization mechanism used in the vector spin models in [20] and [21]. 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.

Extending the Parisi formula along a Hamilton-Jacobi equation

“…The synchronization has been applied in a variety of situations, e.g. [9,4,11], and here we demonstrate another application. A particular synchronization that will be needed here is the one that forces the overlaps µ −1 (α 1 ⋅ α 2 ) and R 1,2 = N −1 σ 1 ⋅ σ 2 to be deterministic functions of their sum in the thermodynamic limit.…”

Free energy in multi-species mixed $p$-spin spherical models

“…The interpolation is reminiscent of [43,Sec. 3] [68], building on the seminal work of Guerra [36] which introduced the technique of RSB interpolation.…”

“…By generalizing this mechanism, Panchenko obtained variational formulas for the free energy of Potts spin glass models [61] and mixed p-spin models with vector spins [60]. The synchronization technique has since been pivotal in a variety of related models [38,27,25,43,52,47]. Using the formula produced by Panchenko in [59], the authors together with Sloman [19] studied symmetry breaking for multi-species SK models (see also [37] from the physics literature).…”

The $\ell^p$-Gaussian-Grothendieck problem with vector spins

“…Lemma 4 in [33] was originally designed to modify the vector spin coordinates in the mixed-pspin model in order to prove the matrix-overlap Ghirlanda-Guerra identities. Using these identities, it is then possible to access the synchronization mechanism [31,32] and find a tight lower bound for the limit of the free energy through the Aizenman-Sims-Starr scheme [22,33]. We will apply this lemma for a different purpose, and, as it turns out, we will need a more explicit expression for the constant L > 0 appearing in the upper bound.…”

“…In addition to proposition 5.4, the proof of theorem 1.2 will rely on the fact that the ℓ p,2 -norm potential in the definition of the Hamiltonian (22) forces the maximizers of this random function to concentrate in a large enough neighbourhood of the origin with overwhelming probability. Lemma 5.5.…”