Mean-Field Spin Glasses and Related Topics

Large complex systems exhibiting a huge number of equilibria and rugged landscapes are important in the mathematical modeling of multiple scientific areas. This project will study certain spin glass models. These models originated in the experimental physics of certain alloys, such as an alloy of gold with a small percentage of iron, possessing unusual magnetic behaviors. This project aims to investigate questions about asymptotic structures and phase transitions of spin glass models that are motivated by emerging topics in high-dimensional optimization, statistical inference, and neural networks. The investigator will mentor and collaborate with graduate students and postdoctoral fellows on this research. The main objective of this work focuses on certain mean-field spin glass models and their diluted variants. The problems include understanding the Gibbs-type variational formulation for the free energy associated with the spiked Gaussian matrix models and concave Hamiltonians. In addition, the asymptotic behavior of the Gaussian operator norm and the limiting free energy in the diluted setting of the Sherrington-Kirkpatrick model and perceptron models will be studied. This research is expected to be beneficial to the many scientific and applied subjects that can be modeled by high complexity randomized combinatorial optimization problems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.