Concentration inequalities

Abstract

We focus on a particularly elegant and effective method, the so-called "entropy method", based on logarithmic Sobolev inequalities and their modifications. Similar ideas appear in a variety of areas of mathematics, including discrete and Gaussian isoperimetric problems, and estimation of mixing times of Markov chains. We intend to shed some light to some of these connections. In particular, we mention some closely related results on influences of variables of Boolean functions, phase transitions, and threshold phenomena.