CAREER: Interacting Particle Systems: Asymptotic Behaviors and Applications

This project studies interacting particle systems at the interface of probability, combinatorics, statistics and applied mathematics. Interacting particle systems naturally emerge in various physical systems, and their behaviors—such as global fluctuation and relaxation rate— are crucial in understanding these systems. Beyond pure mathematics, interacting particle systems have been widely used for random sampling, due to their flexibility and high accuracy.

This project will develop new tools and techniques for advancing the study of interacting particle systems. The project will also provide valuable educational opportunities for students at several levels, including a summer school and workshops focusing on the interactions of probability, mathematical physics, and machine learning theory. These initiatives seek to bring together early-career researchers from diverse fields to foster collaborative efforts.

One primary objective of this project is to investigate the asymptotic behaviors of various interacting particle systems, such as local statistics, global fluctuations and large deviations. These findings will facilitate deriving asymptotic properties of symmetric polynomials, a task less accessible through traditional algebraic combinatorics methods.

Additionally, new universal laws, such as the Tracy-Widom distribution, appear in both random matrix theory and intersecting particle systems. The scaling limits of interacting particle systems converge to two-dimensional universal limiting objects, from which we can recover random matrix eigenvalue statistics by taking one-dimensional slices. Another key aim is to employ tools from interacting particle systems to establish characterizations of random matrix statistics, particularly from the view of line ensembles.

These characterizations provide novel pathways to understand convergence to random matrix statistics in many statistical physics models. Lastly, the PI aims to develop efficient particle-based derivative-free sampling algorithms. These algorithms will enable effective uncertainty quantification for models where differentiation is impractical.

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.