Rare Events and High-Dimensional Stochastic Systems

从物理学和神经科学到工程学和运筹学，许多领域都出现了大量相互作用的随机元素。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，但可以产生重大影响，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。一个重大的数学挑战是了解大量随机元素之间相互作用的结构如何影响这种偏差的性质。这个项目将解决三类随机系统的这一挑战。第一类是由大量相互作用的扩散组成的集合，这些扩散作为金融中的股票价格模型出现，作为生物学和统计物理中的种群动态的连续模型出现。第二类涉及高维度量，例如作为高维数据表示出现的随机矩阵集合，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。理解低维投影的统计和偏差不仅与统计学和数据科学有关，而且对于凸几何中的一些开放猜想也具有重要意义。第三类涉及研究随机矩阵的特征向量的涨落，并涉及与量子力学系统有关的假设。该项目将包括对多个级别的初级研究人员进行纵向综合指导，以及促进扩大任职人数不足和处境不利群体的数学参与的外联工作。这个项目将研究高维随机系统，并致力于描述此类系统中的波动和与平均行为的大偏差以及罕见事件的性质。我们将考虑三类问题。第一个集中于扩散的大集合，它们的局部相互作用结构由潜在的图控制，目的是研究它们的非典型或大偏差行为。近半个世纪以来，在图是完全图的情况下，人们已经很好地理解了这一点，但这个项目的目标是研究当图是(一致)稀疏时的互补情况，这也是许多应用的相关机制。这将需要随机图论、随机分析和变分方法的工具组合。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，主要集中在非对易环境，例如矩阵的Banach空间的范数的水平集，例如p-Schatten空间。文中还将探讨它与凸几何中一些重要猜想的关系。第三个主题涉及从物理学角度研究本征态热化假说的普适性，它是关于随机矩阵的本征向量和相应涨落的陈述。这些问题涉及概率论中的基本问题，并应用于统计物理、渐近凸几何和统计学。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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.