Dynamical and Spatial Asymptotics of Large Disordered Systems

本项目将研究概率论中几种随机模型在长时间动态和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、材料科学、计算机科学和生物学，以及对无序介质中的量子粒子、细菌菌落的生长、交通流量和气体动力学理论等对象的研究。重点是理解普遍性，即微观上不同的概率模型产生相同的极限行为的现象。该项目还包含教育内容，包括课程开发和支持K-12课外数学课程。要研究的具体模型分为三类。第一种是由晶格Schrödinger方程描述的安德森模型，具有i.i.d随机势。主要目标是用数学方法确定局部化现象，即波包不会传播。首席研究员（PI）计划在减少规则假设下对该模型进行全面研究。该项目的第二个主题是kardar - paris - zhang （KPZ）普适性，它描述了各种随机生长过程的尺度极限。在过去的四分之一世纪里，在精确可解结构方面取得了巨大进展。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，包括局部环境极限和大偏差下的缩放极限，以及称为定向景观的极限随机几何。最终目标是将KPZ通用性扩展到精确可解性之外。第三个主题涉及吉布斯采样器，它是用于对高维分布进行采样的蒙特卡洛马尔可夫链（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。一个特殊的例子是Kac从动力学理论出发，其混合时间的顺序是最近几年才确定的。PI计划开发一个总体框架来理解这些吉布斯采样器进化背后的机制，并证明它们的预测截止点。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。

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.