Concentration Phenomena In High Dimensions and Applications to Randomized Models

本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。第一部分研究线性空间上的概率分布在一定凸性假设下的布伦-闵可夫斯基类。它着重于具有重尾的测度的一般无维几何和解析性质，用膨胀、等周和加权sobolev型不等式表示。质量在凸体中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家族。作为一个密切相关的方向，PI还计划考虑大维度积空间集中现象的一些新方面。第二部分讨论了不同浓度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，在最小假设下对观察到的随机变量的依赖性。我们特别感兴趣的是典型分布的渐近行为，它导致给定的随机方案。第三部分致力于分析马尔可夫核的几何特征以及相关的图和其他离散结构上的随机游走。针对马尔可夫半群的收敛速率问题，拟考虑对数Sobolev不等式的离散等周和修正形式。集中现象的研究在很大程度上取决于关于随机过程一般全局性质的各种概率和统计问题。渐近凸几何是另一个领域，高维凸体的许多难题需要集中的结果和技术。本研究还受到组合学和计算机科学（如体积计算）和数学经济学（运输成本优化）中的问题的启发。本研究将继续这一方向的研究，特别是探索弱依赖性在集中现象中的作用，以及它在大维度空间中的适用范围。

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.