Exact Formulas for the KPZ Fixed Point and the Directed Landscape

该项目旨在研究由一大类物理和概率模型产生的极限随机场的性质，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-Parisi-Zhang普适性类，它模拟了许多真实世界的现象，如火灾传播、交通流量或无序的聚合物链。已经推测并部分证明了Kardar-Parisi-Zhang普适类中的所有模型都表现出相同的极限行为。理解这些行为已经成为概率论的一个重要领域，更广泛地说，在数学中也是如此。获奖者是研究生和本科生的导师，从事教育事业。Kardar-Parisi-Zhang普适性类中模型的高度函数预计收敛到一个限制的时空涨落场，称为KPZ不动点。此外，在时空平面上存在一个随机有向度量，它有望支配Kardar-Parisi-Zhang普适性类中的所有模型。这种有向度量称为有向景观。虽然KPZ不动点和有向景观都是Kardar-Parisi-Zhang普适性类研究的中心，但它们直到最近才被刻画出来。该项目旨在使用精确公式的方法来研究这些随机场。研究将首先集中于在Kardar-Parisi-Zhang普适性类的某些精确可解模型中寻找极限场的精确公式，例如定向最后通过渗流；这些公式可用于理解极限场的概率性质。该项目的第二个目标是将上述方法扩展到周期性领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.