Some Interacting Particle Systems and Applications

This project investigates the behavior of large systems of interacting particles over time and space, with a focus on how such systems evolve, stabilize, and occasionally deviate from typical patterns. Such models are foundational to understanding complex dynamics in areas including cloud computing, financial markets, biological synchronization, robotics, and social competition. By developing new mathematical tools to study convergence toward stable configurations and atypical behaviors, this research contributes to core knowledge in probability, statistical physics, and dynamical systems. The project also offers extensive opportunities for research training and workforce development. Together, these efforts contribute to both scientific progress and societal benefits through a deeper understanding of systems central to today’s data-driven and networked environments. Technically, the investigator studies the long-term behavior and scaling limits of several classes of interacting particle systems using mathematical tools such as hydrodynamic limits, fluctuation theory, ergodic analysis, and large deviation principles. The first class of models involves rank-based diffusions arising in stochastic portfolio theory, with emphasis on the infinite Atlas model. The goal is to characterize hydrodynamic limits through Stefan-type free boundary problems, particularly in cases with dense initial configurations. The second set of models originates from load balancing in large-scale service systems, focusing on a regime where server count and traffic intensity are both large. The limiting object in this case is an infinite-dimensional reflected Atlas model, and the study explores its stationary behavior, fluctuation limits, and connection to stochastic partial differential equations. The third model family includes pure jump processes where lagging particles move more frequently, modeling dynamics in distributed computing and financial markets. This research develops hydrodynamic limit theory, studies ergodicity and stability, and investigates the emergence and properties of traveling wave solutions in the associated partial differential equations. Collectively, these studies aim to deepen mathematical understanding of complex, large-scale systems governed by intricate particle interactions. 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.