Exact Formulas for the KPZ Fixed Point and the Directed Landscape

该项目旨在研究从广泛的物理和概率模型中产生的限制随机场的性质，包括随机生长界面，相互作用的粒子系统和随机环境中的聚合物。这个类被称为kardar - paris - zhang通用性类，它模拟了许多现实世界的现象，如火灾传播、交通流量或无序聚合物链。本文推测并部分证明了kardar - paris - zhang普适性类中的所有模型都具有相同的极限行为。理解这些行为已经成为概率论乃至数学中的一个重要领域。该获奖者指导研究生和本科生，并从事教育推广工作。kardar - paris - zhang泛类模型的高度函数被期望收敛于一个极限时空涨落场，这个极限时空涨落场称为KPZ不动点。此外，在时空平面上存在一个随机的有向度规，该度规有望支配kardar - paris - zhang普适性类中的所有模型。这个有向度规叫做有向横向。虽然KPZ固定点和定向景观都是kardar - paris - zhang普世性类研究的核心，但它们直到最近才被描述出来。该项目旨在使用精确公式的方法研究这些随机场。本文将首先对kardar - paris - zhang普适类中某些精确可解模型（如定向末道渗流）的极限场进行精确计算；这些公式可以用来理解极限场的概率性质。这个项目的第二个目标是将上面描述的方法扩展到周期域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.