Concentration of Measure and Functional Inequalities

This thesis is devoted to the study of the concentration of measure phenomenon and its connections with functional inequalities. We focus on the relations between various types of inequalities and in the case of concentration estimates, we are mostly interested in discrete dependent random variables.

In particular, we prove that Beckner inequalities with constants separated from zero as $p\to1^+$ are equivalent to the modified log Sobolev inequality. Further, we derive Sobolev type moment estimates which hold under these functional inequalities. We illustrate these results with applications to concentration of measure estimates for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space.

Then, we answer an open problem posed by Mossel--Oleszkiewicz--Sen regarding relations between $p$-log-Sobolev inequalities for $p\in(0,1]$. We show that for any interval $I\subset (0,1]$, there exist $q,p\in I$, $q