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）资助的。在早期的工作中，PI引入了Stein的可交换对方法的新版本，称为无穷小可交换对，适用于在连续对称群作用下底层随机对象是分布不变的情况。拟开展的项目涉及该新技术的进一步应用，主要集中在两个方面：1)开发一种“高概率”证明某类定量中心极限定理的新方法。例如，大量高维数据点的随机投影“通常”近似为高斯分布；大型wigner型随机矩阵的经验谱测度“通常”接近半圆定律。该项目涉及使用无穷小可交换对的方法与其他理论概率论工具相结合，例如测量浓度和熵界，来证明这类陈述的定量版本。量化这样的陈述不仅可以得到关于收敛性和维度依赖性的更精细的信息，而且还允许在固定（高）维度上应用相关结果，这在几何、统计和计算机科学的应用中通常很重要。2)继续PI对Laplace-Beltrami算子特征函数值分布的研究。在早期的工作中，使用无穷小交换对的方法来识别以前未观察到的特征函数的值分布与其梯度行为之间的联系。在某些例子中，即在大维球面和环面上，得到了部分结果；PI的目标是发展对这些例子的更完整的理解，以及探索以前的结果在固定流形的高特征值极限中的新应用。在过去的四十年里，Stein的方法已经被证明是一个强大的工具，它表明某些随机构造的对象可以通过经典概率分布的近似来很好地理解，并给出了这些近似有多好的定量信息。PI引入了该方法的一个新版本，该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对）。这种新方法已经成功地应用于研究黎曼流形、欧几里得空间中的凸体和紧化的经典矩阵群，在某些情况下证明了与以前已知的结果截然不同的结果，在某些情况下对旧结果有了新的认识。建议的项目包括凸几何，光谱几何，随机矩阵理论和统计学的应用。因此，在拟议的研究中取得的进展将跨越学科，并增加现有的概率技术基础设施。期望在项目过程中开发的新技术将广泛应用于其他数学问题。特别是，这些技术在分析具有两种随机性水平的系统时非常有用，并且系统存在“典型”行为，这种行为的发生取决于其中一种随机性类型的大多数实现；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.