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.
