Exact formulas of the KPZ fixed point and directed landscape and their applications

Many real-world random growth phenomena, such as fire propagation, snow accumulation, traffic flow, and disordered polymer chains, exhibit universal statistical laws. To model these systems, a broad class of processes has been extensively studied, including random growing interfaces, interacting particle systems, and polymers in random environments, and many others. This group of models is called the Kardar–Parisi–Zhang (KPZ) universality class. It is conjectured that all models within this class share identical limiting laws as time approaches infinity. Characterizing these limiting behaviors and establishing this universality has been a fundamental problem in modern probability theory. To advance this field, this project investigates several problems in the KPZ framework while integrating graduate student mentorship and active engagement in educational outreach. The height functions of models within the KPZ universality class are conjectured to converge to a universal space-time field known as the KPZ fixed point. Furthermore, the limiting evolution of these systems is governed by a universal random metric called the directed landscape. This project aims to analyze both the KPZ fixed point and the directed landscape by leveraging the method of exact formulas. To achieve this, the project focuses on two primary objectives: first, to derive novel exact formulas for these fields; and second, to uncover deeper probabilistic properties of these fields using these exact formulas. The project supports the training and professional development of graduate students and junior researchers in this vibrant mathematical direction. 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.