Concentration and Related Probabilistic Phenomena in High Dimensions

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.