Exact Formulas for the KPZ Fixed Point and the Directed Landscape

KPZ 不动点和有向景观的精确公式

• 批准号: 2246683
• 负责人: Zhipeng Liu
• 金额: $ 29.69万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246683&HistoricalAwards=false
• 关键词: Exact; Formulas; KPZ; Fixed; Point

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.

该项目旨在研究从广泛的物理和概率模型中产生的极限随机场的特性，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，它模拟了许多现实世界的现象，如火灾传播、交通流量或无序的聚合物链。本文证明了Kardar-Parisi-Zhang普适类中的所有模型具有相同的极限行为。理解这些行为已经成为概率论和更广泛的数学中的一个重要领域。Kardar-Parisi-Zhang普适性课程中模型的高度函数有望收敛到一个极限时空波动场，即KPZ不动点。此外，在时空平面上存在一个随机的有向度规，它被期望支配Kardar-Parisi-Zhang普适类中的所有模型。这个有向度量称为有向景观。虽然KPZ不动点和有向景观都是研究Kardar-Parisi-Zhang普适性类的核心，但它们的特征只是最近才得到表征。该项目旨在使用精确公式的方法来研究这些随机场。研究将首先集中在寻找Kardar-Parisi-Zhang普适类中某些精确可解模型的极限场的精确公式，例如定向最后一次通过渗流;这些公式可以用来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。