Dynamical and Spatial Asymptotics of Large Disordered Systems

大型无序系统的动力学和空间渐进

• 批准号: 2246664
• 负责人: Lingfu Zhang
• 金额: $ 9.78万
• 依托单位: University of California-Berkeley
• 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=2246664&HistoricalAwards=false
• 关键词: Dynamical; Spatial; Asymptotics; Large; Disordered

This project will study the asymptotic behaviors of several stochastic models in probability theory, in terms of long-time dynamics and static spatial limits. These models find wide applications in various disciplines, including condensed matter physics, material science, computer science, and biology, in the study of objects such as quantum particles in disordered media, the growth of bacterial colonies, traffic flow, and the kinetic theory of gases. A focus is to understand universality, the phenomenon where microscopically different probabilistic models produce the same limiting behavior. This project also contains educational components, including curriculum development and supporting K-12 extracurricular math programs.The specific models to be investigated fall into three categories. The first is the Anderson model described by the lattice Schrödinger equation with i.i.d. random potentials. The main objective is to mathematically establish the localization phenomenon, where wave packets do not spread. The principal investigator (PI) plans to carry out comprehensive studies of this model under reduced regularity assumptions. The second theme of this project is the Kardar-Parisi-Zhang (KPZ) universality, which describes the scaling limit of various random growth processes. In the past quarter-century, enormous progress has been made on those with exact-solvable structures. The PI will use geometric and probabilistic methods to study the asymptotics of several such exactly-solvable models, including local environment limits and scaling limits under large deviation, and a limiting random geometry termed the directed landscape. The ultimate goal is to extend KPZ universality beyond exact-solvability. The third topic concerns Gibbs samplers, which are Monte Carlo Markov Chain (MCMC) algorithms used to sample high-dimensional distributions. The focus is on the continuous state space setting, where tools to analyze time evolution are relatively limited. A particular instance is Kac's walk from kinetic theory, whose order of mixing time was only determined in recent years. The PI plans to develop a general framework to understand the mechanism behind the evolution of these Gibbs samplers, and prove predicted cutoffs for them.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.

本计画将研究机率论中几种随机模型在长时间动态与静态空间限制下的渐近行为。这些模型广泛应用于各个学科，包括凝聚态物理学，材料科学，计算机科学和生物学，研究对象，如无序介质中的量子粒子，细菌菌落的生长，交通流量和气体的动力学理论。一个重点是理解普遍性，微观上不同的概率模型产生相同的限制行为的现象。该项目还包含教育部分，包括课程开发和支持K-12课外数学计划。第一种是由具有i.i.d.的格点薛定谔方程描述的安德森模型。随机势主要目的是建立数学上的本地化现象，波包不扩散。主要研究者（PI）计划在简化的规律性假设下对该模型进行全面研究。这个项目的第二个主题是Kardar-Parisi-Zhang（KPZ）普适性，它描述了各种随机增长过程的标度极限。在过去的四分之一世纪里，在具有精确可解结构的问题上取得了巨大的进展。PI将使用几何和概率方法来研究几个这样的精确可解模型的渐近性，包括局部环境限制和大偏差下的缩放限制，以及称为定向景观的限制随机几何。最终目标是将KPZ的普适性扩展到精确可解性之外。第三个主题涉及Gibbs采样器，它是用于对高维分布进行采样的蒙特卡罗马尔可夫链（MCMC）算法。重点是在连续状态空间设置，工具来分析时间演化是相对有限的。一个特殊的例子是卡茨从动力学理论出发的步行，其混合时间的顺序只是在最近几年才确定的。PI计划开发一个通用框架，以了解这些吉布斯采样器的演变背后的机制，并证明他们预测的截止值。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。