Kardar-Parisi-Zhang Universality Class, Integrable Differential Equations, and Spin Glass

这个项目是关于一个大型概率系统中成分之间有时复杂和随机的相互作用如何导致简单和普遍的行为。在这里，普适性意味着这些性质是针对一大类这样的系统而获得的。我们将研究几类具体的概率模型及其基本性质，特别是当系统规模变大时。这个项目将有助于提高我们对概率模型普适性的范围和局限性的理解。该项目的许多部分将与研究生和年轻研究人员一起开展，帮助他们的专业发展。具体地说，将研究Kardar-Parisi-Zhang(KPZ)普适类模型以及它们与可积微分方程和自旋玻璃模型的联系。特别令人感兴趣的是KPZ普适性类模型在环域上的多时间分布，以了解和拓宽KPZ模型与可积微分方程之间的关系，并研究受确定性场扰动的球形Sherrington-Kirkpatrick模型的涨落。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。

This project is about how sometimes complex and random interactions between the constituents in a large probabilistic system may lead to simple and universal behaviors. Here universality means that these properties are attained for a large class of such systems. A few concrete classes of probabilistic models and their fundamental properties will be investigated, especially when the system sizes become large. This project will help improve our understanding of the scopes and limitations of the universality of probability models. Many parts of the project will be carried out with graduate students and young researchers, helping their professional development. In concrete terms, the Kardar-Parisi-Zhang (KPZ) universality class models will be studied as well as their connections to integrable differential equations, and spin glass models. Of particular interest will be the multi-time distributions of KPZ universality class models on the ring domain, to understand and broaden the relations between the KPZ models and integrable differential equations, and to study the fluctuations of the spherical Sherrington-Kirkpatrick model perturbed by a deterministic field.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.