Concentration and Related Probabilistic Phenomena in High Dimensions

这个项目处理了在过去十年中遇到的几个关于高维的具有挑战性的问题，主要是概率内容。这些问题包括满足无维集中性质的概率分布的表征问题和相关的sobolev型不等式，关于线性泛函相对于对数凹测度的主导作用的kls猜想，随机求和模型中典型分布的存在性和渐近正态性问题。PI还计划在其他对称假设下解决乘积测量的集中现象的新方面。与此密切相关的是，这项活动的一部分致力于乘积空间上排列不变概率测度的有限维de Finetti表示及其在量化长有限可交换序列中元素的依赖率方面的应用。一个单独的部分致力于分析离散结构上的随机游走，并着重于开发基于对数Sobolev不等式的适当修改形式的技术。集中现象的研究主要是由关于随机过程光滑泛函的一般全局性质的经典概率和统计问题引起的。集中工具在渐近凸几何中也非常重要，在渐近凸几何中，人们探索高维凸物体的尺寸的作用。由于应用的普遍性，这一领域的研究已经变得相当丰富，并被证明是有用的；另一方面，它积累了许多基本的开放性问题，吸引了许多研究者。本研究旨在推动多维现象的研究，探索其与广义弱依赖相关的重要效应之间的联系。

0405587 Bobkov This project deals with several challenging problems about high dimensions, mainly of probabilistic content, that have been encountered in the last decade. These include the problem of characterization of probability distributions satisfying dimension free concentration properties and related Sobolev-type inequalities, the KLS-conjecture on the dominating role of linear functionals with respect to logarithmically concave measures, the problem of existence and asymptotic normality of typical distributions in randomized models of summation. The PI also plans to address new aspects of the concentration phenomenon of product measures under additional symmetry hypotheses. As closely related, part of this activity is devoted to finite dimensional de Finetti representations for permutation invariant probability measures on product spaces and their applications to quantifying the rate of dependence of elements in long finite exchangeable sequences. A separate part is devoted to analysis of random walks on discrete structures and focuses on developing techniques based on suitable modified forms of logarithmic Sobolev inequalities. The study of concentration phenomena is motivated, in particular, by classical problems of probability and statistics about general global properties of smooth functionals of stochastic processes. Concentration tools are also of great importance in asymptotic convex geometry where one explores the role of the dimension of high-dimensional convex objects. This area of research has become rather rich and proved to be useful due to the universal character of applications; on the other hand, it has accumulated a number of fundamental open questions attracting many investigators. The proposed research is aimed to push forward the study of multidimensional phenomena and to explore their connections with important effects related to the weak dependence in a broad sense.