Statistics of metastable states in the spherical models and beyond

Abstract

Characterizing metastable states and saddle points in complex systems can shed light on a variety of equilibrium and out-of-equilibrium behaviors. In this talk, I will discuss the most versatile theoretical technique for this, the Kac–Rice method, and its application to a family of mean-field models, the spherical spin glasses. I will draw connections from this technique to random matrix theory and Faddeev–Popov gauge fixing. For the spherical models, writing down an effective action is straightforward but finding physical saddle points can be challenging. I will show how, for the lowest-energy states, a BRST supersymmetry can be used to simplify the action and map it to a more tractable equilibrium problem. Finally, I will discuss challenges to using these techniques in inference problems with non-Gaussian energy functions, and current work being done to address them.