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.

许多领域都会出现大量相互作用的随机元素，从物理和神经科学到工程和操作研究。研究波动和与这些系统的典型或平均行为的巨大偏差非常重要。实际上，波动和非典型事件虽然很少见，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情况很重要。 一个重大的数学挑战是查看大型随机元素之间相互作用的结构如何影响这种偏差的性质。该项目将针对三类随机系统解决这一挑战。头等舱由大量的相互作用扩散集成，这些扩散是金融中股票价格的模型，作为生物学中人口动态的连续性模型以及统计物理学的连续性模型。 第二类涉及高维度的衡量标准，例如作为高维数据表示的矩阵的随机组合及其与较低维投影的关系，在分析数据时，它们用作减少维度的技术。 了解低维投影的统计和偏差不仅与统计和数据科学有关，而且对凸几何的某些开放猜想具有重要意义。 第三类涉及随机矩阵特征向量的波动的研究，并解决了与量子机械系统有关的假设。该项目将包括在多个层面上垂直整合初级研究人员的指导，以及促进扩大代表性不足和处于弱势群体的数学参与的努力。该项目将研究高维的随机系统，并努力表征与平均行为的波动和巨大偏差，以及此类系统中罕见事件的性质。将考虑三类问题。第一个侧重于大量扩散集合，其局部相互作用结构受基本图的控制，旨在研究其非典型或大偏差行为。 尽管在底层图是完整图的情况下，近半个世纪以来，近半个世纪以来，这是对此的良好理解，但该项目的目的是研究图形（均匀）稀疏的互补情况，这也是许多应用程序的相关制度。这将需要从随机图理论，随机分析和变异方法中使用工具的组合。第二类问题与高维凸体投影的浓度和较大偏差行为有关，重点是非共同环境，例如矩阵的Banach空间的水平集，例如P-Schatten空间。还将探索与凸几何形状中一些出色猜想的关系。第三个主题涉及研究物理学特征态热假说的普遍性，该假设是关于随机矩阵的特征向量和相应波动的陈述。 这些问题解决了概率理论中的基本问题，并在统计物理学，渐近凸几何和统计中有应用。该项目将采用分析，几何和概率方法的结合。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。