Periodic Kardar-Parisi-Zhang (KPZ) Universality

许多随机生长模型，如火的传播或细菌菌落的生长，被认为具有一定的普遍规律。分析这种模式的数学机制是近二十年来一个活跃的研究领域。由于概率论和数理物理界的突破性进展，越来越多的模型被成功地分析，并发现它们具有相同的大时间限制行为。这些模型被称为属于Kardar- paris -Zhang （KPZ）通称类，以Mehran Kardar、Giorgio Parisi和Yi-Cheng Zhang的名字命名，他们引入了一个非线性随机偏微分方程，即所谓的KPZ方程，来描述随机生长界面。本课题旨在研究具有空间周期性的KPZ通用性类模型。本课题的目标之一是分析周期可解增长模型，并通过探索与这些模型相关的贝特根的结构和渐近性来理解它们在不同参数尺度下的普遍极限行为。另一个目标是通过获得周期变得足够大时周期可解增长模型的精确结果，开辟一个新的方向来接近无限空间中的KPZ普适性类。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。

Many random growth models, such as fire propagation or bacterial colony growth are believed to share certain universal pattern. Analyzing the mathematical mechanism of such pattern has been an active research area in the last twenty years. Due to the breakthrough progress in the probability and mathematical physics community, an increasing number of models have been successfully analyzed and they are found to share the same large time limiting behaviors. These models are called to belong to the Kardar-Parisi-Zhang (KPZ) universality class, named after Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang who introduced a non-linear stochastic partial differential equation, the so-called KPZ equation, to describe the random growing interfaces. This project aims to study models in the KPZ universality class with spatial periodicity. One goal of this project is to analyze the periodic solvable growth models and understand their universal limiting behaviors under different parameter scales by probing the structure and asymptotics of the Bethe roots associated with these models. The other goal is to develop a new direction to approach the KPZ universality class in the infinite space by obtaining exact results of periodic solvable growth models when the period becomes sufficiently large.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.