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.

该项目将研究概率论中几种随机模型在长期动力学和静态空间限制方面的渐近行为，这些模型在凝聚态物理、材料科学、计算机科学和生物学等各个学科中都有广泛的应用。在无序介质中的量子粒子、细菌菌落的生长、交通流和气体动力学理论等物体的研究中，重点是理解普遍性，即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 课外数学项目。要研究的具体模型分为三类，其主要目标是由具有独立同分布的格子薛定谔方程描述的。该项目的第二个主题是在数学上建立波包不扩散的局域化现象。 Kardar-Parisi-Zhang (KPZ) 普适性，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中，PI 将使用几何和概率方法来解决那些具有精确可解结构的问题。研究几个此类精确可解模型的渐近性，包括大偏差下的局部环境限制和缩放限制，以及称为有向景观的限制随机几何。最终目标是将 KPZ 普遍性扩展到超越。第三个主题涉及吉布斯采样器，它是用于采样高维分布的蒙特卡罗马尔可夫链（MCMC）算法，重点是连续状态空间设置，其中分析时间演化的工具相对有限。特例是 Kac 从动力学理论出发，其混合时间顺序是近年来才确定的。PI 计划开发一个通用框架来了解这些吉布斯采样器演化背后的机制，并证明它们的预测截止值。授予 NSF 的法定使命，并通过评估反映使用基金会的智力优点和更广泛的影响审查标准，被认为值得支持。