Interacting Particle Systems and Beyond

The project focuses on the study of lattice models in statistical mechanics. The algebraic structure inherent in such models allows for many exact computations, while their probabilistic nature provides a new point of view and interpretation of the underlying algebraic data. For instance, these systems can be used to study how crystals melt, how neurons move through the brain, how a fire front advances, how a cancer spreads, how a plankton colony grows in the ocean. The research project aims at achieving a better understanding of the macroscopic behavior of some important models as the size of the system grows. The models under consideration are usually referred to as “integrable” or “exactly solvable.” Though exactly solvable systems are very special, their asymptotic properties are believed to be representative for a larger family of models. In this way, besides being interesting themselves, exactly solvable systems are exemplars of their conjectured universality classes and can be used to build intuition and make predictions. The aim is to obtain a variety of robust methods to study the universality classes. This research program seeks to establish a better understanding of some “universality” features in the context of certain interacting particle systems. The research program consists of three main directions— the study of log-gases (ensemble of particles on the line confined by an external potential that repel each other logarithmically), the study of the stationary measures for the Kardar-Parisi-Zhang equation (a non-linear stochastic partial differential equation that was originally proposed as a model of surface growth), and the study of traffic models. 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.