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 se