Dynamics of Lattice and Mean-Field Spin Systems

Dynamical evolutions in complex high-dimensional energy landscapes are ubiquitous in modern science. Prototypical examples include protein folding, evolutionary dynamics, out-of-equilibrium thermodynamics, and training of neural networks. This project will investigate these dynamical evolutions by analyzing Markov chains arising from spin systems in statistical physics. These models are both central to modern probability theory and have implications for other fields, including theoretical computer science, high-dimensional statistics, and machine learning. The project also includes broadening participation efforts at Northwestern University and mathematical education outreach to local K-12 schools and correctional institutions. This project is focused on Markov chain evolutions in three important settings: (1) spin systems (e.g., the Ising and Potts models) on lattices, (2) random surfaces and interfaces between thermodynamically stable phases, and (3) disordered mean-field models (e.g., spin glasses) and loss functions in high dimensions. These diverse set of dynamical processes are expected to exhibit a wide range of overlapping mathematical and physical phenomena. We study them with an eye towards understanding the mechanisms dictating convergence times to equilibrium, especially the roles played by varying the boundary conditions, initializations, and restrictions to the state space. 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.