In statistical physics, the study of spin glasses was initialized to describe the low temperature state of a class of magnetic alloys in the 1960s. The Sherrington-Kirkpatrick (SK) model is a mean field approximation of the physical short range spin glass model introduced in the 1970s. Starting in 1979, Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model. Since then, his method has been applied to study various complex systems, which eventually earned him the 2021 Nobel Prize in Physics.

  In this talk, I will first introduce Parisi's conjectures and rigorous results by Talagrand and Panchenko. Then I will show that his prediction on infinite replica symmetry breaking holds at zero temperature for the more general mixed p-spin model. As an example for the application of Parisi's method, I will present Fyodorov and Le Doussal's prediciton on the Hessian spectrum at the global minimum of locally isotropic Gaussian random fields. A partial solution will be provided via landscape complexity.

About the reporter: Zeng Qiang graduated from Beijing Normal University with a bachelor's degree, Peking University with a master's degree, and the University of Illinois at Champaign with a doctor's degree. He later engaged in postdoctoral research at Harvard University, Berkeley Institute of Mathematical Sciences, and Northwestern University. Now he is an associate researcher of the Institute of Mathematics and Systems Science of the Chinese Academy of Sciences. The research field is non commutative probability and spin glass, and a series of works have been done on non commutative martingale and concentration inequality, replica symmetry breaking of spin glass, and landscape complexity. The relevant results have been published in international high-level journals such as CPAM, CMP, AOP, PTRF, etc.