## Departmental PhD Thesis Exam – Justin Ko

Wednesday, August 25, 2020
11:00 a.m.

PhD Candidate:  Justin Ko
Supervisor:   Dmitry Panchenko
Thesis title: The Free Energy of Spherical Vector Spin Glasses

We study a class of vector spin models with configurations restricted to subsets of the sphere. We will prove a constrained free energy formula for these models. This formula defines a large deviations principle for the limiting distribution of the overlaps under the asymptotic Gibbs measure. The thesis builds on the mathematical results used to prove free energy formulas for the classical Sherrington–Kirkpatrick spin glass, spherical spin models, and vector spin glass models. The free energy formula proved in this thesis are true generalizations of the classical results, in the sense that these vector spin formulas restricted to one dimension coincide with the known results for classical models.

The first contribution of this thesis is a variational formula for contrained copies of classical spherical spin glasses sampled at different temperatures. The free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied by Panchenko and Talagrand. They proved an upper bound of the constrained free energy using Guerra’s interpolation. In this thesis, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman–Sims–Starr scheme and the synchronization mechanism used in the vector spin models. 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.

The second contribution of this thesis is the simplification of this variational formula to the form originally discovered for the classical spherical spin glass model by Crisanti and Sommers. In particular, we prove the analogue of the Crisanti–Sommers variational formula for spherical spin glasses with vector spins. This formula is derived from the discrete Parisi variational formula for the limit of the free energy of constrained copies of spherical spin glasses. In vector spin models, the variations of the functional order parameters must preserve the monotonicity of matrix paths which introduces a new challenge in contrast to the derivation of the classical Crisanti–Sommers formula.

A copy of the thesis can be found here: ut-thesis-Ko-updated