Exact Formulas for the KPZ Fixed Point and the Directed Landscape

The project aims to study the properties of limiting random fields arising from a broad class of physical and probabilistic models, including random growing interfaces, interacting particle systems, and polymers in random environments. This class is called the Kardar-Parisi-Zhang universality class, and it models many real-world phenomena, such as fire propagation, traffic flow, or disordered polymer chains. It has been conjectured and partially proved that all models in the Kardar-Parisi-Zhang universality class exhibit the same limiting behaviors. Understanding these behaviors has become an important area in probability theory and more generally in mathematics. The awardee mentors graduate and undergraduate students and is engaged in educational outreach. The height functions of models in the Kardar-Parisi-Zhang universality class are expected to converge to a limiting space-time fluctuation field, which is called the KPZ fixed point. Moreover, there is a random directed metric on the space-time plane that is expected to govern all the models in the Kardar-Parisi-Zhang universality class. This directed metric is called the directed landscape. While both the KPZ fixed point and the directed landscape are central to the study of the Kardar-Parisi-Zhang universality class, they have only been characterized very recently. The project aims to study these random fields using the approach of exact formulas. The research will first focus on finding exact formulas for the limiting fields in certain exactly solvable models in the Kardar-Parisi-Zhang universality class, such as the directed last passage percolation; these formulas can be used to understand probabilistic properties of the limiting fields. A second goal of this project is to extend the approaches described above to periodic domains. 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.