Concentration Phenomena In High Dimensions and Applications to Randomized Models

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 the

Brunn-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.