Classical concentration inequalities allow to control deviation of a Lipschitz function from its mean. In this talk I shall present a method based on functional inequalities which allows to handle the class of smooth functions with bounded derivatives of higher order (so, in particular, multivariate polynomials). When combined with Latała’s estimates for Gaussian chaoses, it leads to exponential-type tail estimates for smooth functions in random vectors which satisfy e.g. the logarithmic Sobolev inequality. I will also present related estimates for polynomials in independent sub-Gaussian random variables. Applications to linear eigenvalue statistics of random matrices and a problem of subgraph counts in random graphs will also be discussed.
Based on joint work with R. Adamczak (University of Warsaw).