Concentration Phenomena In High Dimensions and Applications to Randomized Models

该项目解决了与集中现象相关的高维概率对象的几个具有挑战性的问题。第一部分致力于研究线性空间上的概率分布在一定的凸性假设下的thebrunn-Minkowski类。它侧重于一般的无量纲的几何和分析性质的措施与重尾，表示在膨胀，等周，加权Sobolev型不等式。凸体中质量的均匀分布和更一般的对数凹分布描述了凸或双曲测度层次中的另一个重要家族。作为一个密切相关的方向，PI也计划考虑一些新的方面的集中现象的产品空间的大尺寸。第二部分讨论了不同的集中现象在随机模型中的应用，如数据的部分和，随机矩阵的谱等，在最小的假设下，对所观察到的随机变量的依赖性。一个特别感兴趣的是一个典型的分布，导致在一个给定的随机方案的渐近行为。第三部分是关于图和其他离散结构上的马尔可夫核和相关随机游动的几何特征的分析。离散等周和修改形式的对数Sobolev不等式计划被认为是与问题的收敛速度的马尔可夫半群。浓度现象的研究是强烈的各种问题的概率和统计的一般全局性质的随机过程。渐近凸几何是另一个领域，其中许多关于高维凸体的困难问题都需要集中的结果和技巧。本研究还受到组合数学和计算机科学（如体积计算）和数学经济学（运输成本优化）问题的启发。目前的建议继续在这个方向上的研究，特别是旨在探索浓度现象中的弱依赖性的作用，以及它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.