Geometry of spherical spin glasses

Spherical spin glasses are canonical models for smooth random functions in high dimensions. In this review, we survey several interrelated lines of research on their geometric structure. We begin with results concerning critical points and their relationship to the Gibbs measure. For the pure models, the measure concentrates on spherical bands around critical points that approximately maximize the energy at a particular radius. Next, we present another approach in which a similar picture is derived for general mixed models. At the core of this approach is a free energy functional computed over bands using multiple orthogonal replicas, satisfying a strong concentration of measure. We discuss several implications of this method for a generalized Thouless-Anderson-Palmer (TAP) approach. Finally, we explain how these geometric insights inform optimization algorithms, and briefly relate them to Smale's 17th problem over the real numbers.