Concentration Phenomena In High Dimensions and Applications to Randomized Models

这个项目解决了几个关于高维概率对象的挑战性问题，这些问题与集中现象有关。第一部分主要研究在Brunn-Minkowski类的某些凸性假设下线性空间上的概率分布。它主要讨论了重尾测度的一般无因次几何性质和解析性质，这些性质用膨胀、等周和加权的Sobolev型不等式表示。凸体中质量的均匀分布和更一般的对数凹分布描述了凸度量或双曲度量层次中的另一个重要的族。作为一个密切相关的方向，PI还计划考虑大维度乘积空间上的集中现象的一些新方面。第二部分讨论了不同的集中现象在随机模型中的应用，如数据的部分求和、随机矩阵谱等，在最小假设下的随机变量的相依性。特别感兴趣的是典型分布的渐近行为，导致给定的随机化方案。第三部分分析了马尔可夫核的几何特征及其在图和其他离散结构上的随机游动。关于马尔可夫半群的收敛速度问题，我们计划考虑离散等周形式和修正形式的对数Sobolev不等式。集中现象的研究强烈地取决于概率统计关于随机过程的一般全局性质的各种问题。渐近凸几何是另一个领域，高维凸体的一些困难问题吸引着集中结果和技巧。这项研究还受到组合学和计算机科学问题(如运量的计算)和数理经济学问题(运输成本的优化)的推动。本提案继续这方面的研究，特别是旨在探讨弱依赖在集中现象中的作用及其在大尺寸空间中的适用范围。

This project addresses several challenging problems about high-dimensional probabilistic objects, that are related to the concentration phenomena. Part one is devoted to the study of probability distributions on linear spaces under certain convexity hypotheses of theBrunn-Minkowski kind. It focuses on general dimension free geometric and analytic properties of measures with heavy tails, expressed in terms of dilation, isoperimetric, and weighted Sobolev-type inequalities. The uniform distribution of mass in a convex body and more general log-concave distributions describe another important family in the hierarchy of convex or hyperbolic measures. As a closely related direction, the PI is also planning to consider some new aspects of the concentration phenomenon on product spaces of a large dimension. Part two deals with applications of different concentration phenomena to the randomized models, such as a partial summation of data, spectrum of stochastic matrices, etc., under minimal assumptions on the dependence of the observed random variables. Of a particular interest is an asymptotic behavior of typical distributions, resulting in a given randomized scheme. Part three is devoted to analysis of geometric characteristics of Markov kernels and associated random walks on graphs and other discrete structures. Discrete isoperimetric and modified forms of logarithmic Sobolev inequalities are planned to be considered in connection with the problem on the rates of the convergence of the Markov semi-groups.The study of the concentration phenomena is strongly dictated by various problems of Probability and Statistics on general global properties of stochastic processes. The Asymptotic Convex Geometry is another field, where a number of hard problems about high-dimensional convex bodies appeal to the concentration results and techniques. This study is also stimulated by problems in Combinatorics and Computer Science (such as computation of the volume) and in Mathematical Economics (optimization of the transport costs). The present proposal continues research in this direction and is aimed, in particular, to explore the role of the weak dependence in concentration phenomena, as well as its range of applicability in spaces of large dimensions.