Applications of the method of infinitesimal exchangeable pairs in analysis, geometry and statistics

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。在早期的工作中，PI引入了Stein的可交换对方法的一个新版本，称为无穷小可交换对，适用于在连续对称群的作用下潜在随机对象的分布不变的情况。建议的项目涉及这一新技术的进一步应用，主要集中在两个主要方向：1)发展一种新的方法来证明某种类型的“高概率”的量化中心极限定理。例如，大量高维数据点的随机投影“通常”近似为高斯；大型维格纳型随机矩阵的经验谱度量“通常”接近半圆定律。拟议的项目涉及使用无穷小可交换对的方法结合其他理论概率工具，例如测量集中度和熵界，来证明这类陈述的量化版本。量化这些陈述不仅可以得到关于收敛和维度依赖的更精细的信息，而且还允许将相关结果应用于固定(高)维，这在几何、统计学和计算机科学的应用中经常是重要的。2)继续PI对Laplace-Beltrami算子的本征函数的值分布的研究。在早期的工作中，使用极小可交换对的方法来确定特征函数的值分布与其梯度行为之间以前未观察到的联系。在某些例子中，即在高维球面和环面上，已经获得了部分结果；PI旨在更全面地理解这些例子，并探索先前结果在固定流形上的高特征值极限方面的新应用。在过去的40年里，Stein的方法被证明是一个强大的工具，可以证明某些随机构造的物体可以通过经典概率分布的逼近来很好地理解，并给出关于这些逼近有多好的定量信息。PI引入了一种新版本的方法，可用于利用问题中存在的“连续对称性”(例如，球体的对称性与立方体的对称性相反)。这种新的方法已经成功地应用于黎曼流形、欧氏空间中的凸体和紧致经典矩阵群的研究，在某些情况下证明了与已知结果相当不同的结果，在某些情况下为旧结果提供了新的线索。拟议的项目包括在凸几何、谱几何、随机矩阵理论和统计学中的应用。因此，拟议中的研究取得的进展将跨越多个学科，并增加现有的概率技术基础设施。预计在项目过程中开发的新技术将广泛适用于数学中的其他问题。具体地说，这些技术在分析存在两种随机性的系统中可能是有用的，并且系统存在一种“典型”行为，其发生条件是其中一种随机性的大多数实现；这种情况在数学和统计物理中经常发生。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). In earlier work, the PI introduced a new version of Stein's method of exchangeable pairs, called infinitesimal exchangeable pairs, adapted to situations in which the underlying random object is distributionally invariant under the action of a continuous symmetry group. The proposed project involves further applications of this new technique, focusing on two main directions:1) Developing a new approach to proving a certain type of quantitative central limit theorem ``with high probability''. For example, a random projection of a large collection of high-dimensional data points is ``usually'' approximately Gaussian; the empirical spectral measure of a large Wigner-type random matrix is ``usually'' close to the semi-circle law. The proposed project involves using the method of infinitesimal exchangeable pairs in combination with other tools of theoretical probability, e.g. measure concentration and entropy bounds, to prove quantitative versions of statements of this type. Quantifying such statements leads not only to finer information about convergence and dimensional dependence, but also allows applications of the relevant results in fixed (high) dimensions, which is frequently important in applications to geometry, statistics, and computer science.2) Continue the PI's study of value distributions of eigenfunctions of the Laplace-Beltrami operator. In earlier work, the method of infitesimal exchangeable pairs was used to identify a previously unobserved connection between value distributions of eigenfunctions and the behavior of their gradients. Partial results have been obtained in certain examples, namely on large-dimensional spheres and tori; the PI aims to develop a more complete understanding of these examples as well as exploring new applications of previous results in the high-eigenvalue limit on fixed manifolds.Over the past four decades, Stein's method has proved to be a powerful tool for showing that certain randomly constructed objects can be well understood by approximating by classical probability distributions, and giving quantitative information about how good these approximations are. The PI has introduced a new version of the method which can be used to take advantage of the presence of "continuous symmetries" (e.g., the symmetries of the sphere as opposed to those of the cube) in a problem. This new approach has already been successfully applied in studying Riemannian manifolds, convex bodies in Euclidean space, and the compact classical matrix groups, in some cases to prove results rather different from those previously known, and in some cases shedding new light on old results. The proposed project includes applications in convex geometry, spectral geometry, random matrix theory, and statistics. Accordingly, progress made in the proposed research will cut across disciplines as well as adding to the existing infrastructure of available techniques in probability. It is expected that the new techniques developed in the course of the project will be widely applicable to other problems in mathematics. In particular, these techniques will likely be useful in analyzing systems in which there are two levels of randomness, and there is a "typical" behavior for the system, which occurs conditioned on most realizations of one of the types of randomness; such situations occur frequently in mathematics and statistical physics.