Rare Events and High-Dimensional Stochastic Systems

稀有事件和高维随机系统

• 批准号: 2246838
• 负责人: Kavita Ramanan
• 金额: $ 36.5万
• 依托单位: Brown University
• 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: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246838&HistoricalAwards=false
• 关键词: Rare; Events; Dimensional; Stochastic; Systems

Large collections of interacting random elements arise in many areas, ranging from physics and neuroscience to engineering and operations research. It is of great importance to study fluctuations and large deviations from the typical or mean behavior of these systems. Indeed, fluctuations and atypical events, although rare, can have significant impact, so it is important to quantify these probabilities and to understand typical occurrences of rare events. A significant mathematical challenge is to see how the structure of interaction between large collections of stochastic elements influences the nature of such deviations. This project will address this challenge for three classes of stochastic systems. The first class consists of large collections of interacting diffusions that arise as models of stock prices in finance, as continuum models of population dynamics in biology, and in statistical physics. The second class concerns high-dimensional measures, such as random ensembles of matrices arising as representations of high-dimensional data, and their relation to lower-dimensional projections, which are used as a dimension-reduction technique when analyzing data. Understanding the statistics and deviations of lower-dimensional projections is not only relevant for statistics and data science but also has significance for some open conjectures in convex geometry. The third class pertains to the study of fluctuations of eigenvectors of random matrices and addresses hypotheses related to quantum mechanical systems. The project will include vertically integrated mentoring of junior researchers at multiple levels and outreach efforts to foster broadening the mathematical participation of underrepresented and disadvantaged groups. This project will study high-dimensional stochastic systems and work to characterize fluctuations and large deviations from mean behavior and the nature of rare events in such systems. Three classes of problems will be considered. The first focuses on large collections of diffusions whose local interaction structure is governed by an underlying graph, and aims to study their atypical or large deviation behavior. While this has been well understood for almost half a century in the case when the underlying graph is the complete graph, the goal of this project is to study the complementary case when the graphs are (uniformly) sparse, which is also the relevant regime for many applications. This will require a combination of tools from random graph theory, stochastic analysis and variational methods. The second class of problems relates to the study of concentration and large deviation behavior of projections of high-dimensional convex bodies, with a focus on non-commutative settings, such as the level sets of norms of Banach spaces of matrices such as the p-Schatten spaces. The relation to some outstanding conjectures in convex geometry will also be explored. The third theme concerns the study of universality of the Eigenstate Thermalisation Hypothesis from physics, which is a statement about eigenvectors of random matrices, and corresponding fluctuations. These problems address fundamental problems in probability theory and have applications to statistical physics, asymptotic convex geometry and statistics. The project will employ a combination of analytical, geometric and probabilistic methods.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.

大量的相互作用的随机元素出现在许多领域，从物理学和神经科学到工程和运筹学。研究这些系统的波动和偏离典型或平均行为的大偏差是非常重要的。事实上，波动和非典型事件虽然罕见，但可能会产生重大影响，因此量化这些概率并了解罕见事件的典型发生是很重要的。 一个重要的数学挑战是，看看大的随机元素集合之间的相互作用的结构如何影响这种偏差的性质。这个项目将解决这三个类的随机系统的挑战。第一类由大量的相互作用扩散组成，这些相互作用扩散在金融学中作为股票价格模型出现，在生物学中作为种群动力学的连续模型出现，在统计物理学中作为连续模型出现。 第二类涉及高维度量，例如作为高维数据表示而产生的矩阵的随机集合，以及它们与低维投影的关系，这在分析数据时用作降维技术。 了解低维投影的统计和偏差不仅与统计学和数据科学有关，而且对凸几何中的一些开放式几何也有重要意义。 第三类涉及随机矩阵的本征向量的波动的研究，并解决与量子力学系统有关的假设。该项目将包括在多个层面上对初级研究人员进行纵向一体化的指导，并开展外联工作，以促进扩大代表性不足和弱势群体的数学参与。本项目将研究高维随机系统，并致力于描述波动和偏离平均行为的大偏差以及此类系统中罕见事件的性质。将考虑三类问题。第一个集中在大型集合的扩散，其局部相互作用的结构是由一个基本的图形，并旨在研究他们的非典型或大偏差的行为。 虽然这已经很好地理解了几乎半个世纪的情况下，当底层图是完整的图，这个项目的目标是研究互补的情况下，图是（均匀）稀疏，这也是许多应用程序的相关制度。这将需要从随机图论，随机分析和变分方法的工具的组合。第二类问题涉及高维凸体投影的集中和大偏差行为的研究，重点是非交换设置，例如矩阵的Banach空间（如p-Schatten空间）的范数的水平集。在凸几何中的一些杰出的代数的关系也将被探讨。第三个主题是研究物理学中本征态热化假说的普适性，这是一个关于随机矩阵的本征向量和相应涨落的陈述。 这些问题解决了概率论中的基本问题，并应用于统计物理，渐近凸几何和统计。该项目将采用分析、几何和概率方法相结合的方法。该奖项反映了NSF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。