Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Abstract

We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as p → 1 + are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities.We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space. Contents1. Introduction 1.1. Motivation and informal presentation 1.2. General setting 1.3. Examples 1.4. Functional inequalities 2. From modified log-Sobolev to Beckner's inequalities 2.1. Main result 2.2. Auxiliary lemmas 2.3. Proof of Theorem 2.1 3. Moment estimates derived from Beckner's inequalities 4. Applications 4.1. The continuous setting 4.2. Product spaces 4.3. Glauber dynamics 4.4. The symmetric group 4.5. Stochastic covering property 4.6. Zero-range processes 4.7. The Poisson space 5. Higher order concentration 5.1. Abstract inequality 5.2. Discussion on the choice of gradients 5.3. Applications to tetrahedral polynomials Appendix A.

Concentration on Poisson spaces via modified $Φ$-Sobolev inequalities

“…Independently and, in a sense, in parallel with us Adamczak, Polaczyk and Strzelecki [1] have very recently obtained (among many other results) moment estimates and concentration bounds for Poisson functionals, which are similar to the ones we prove, see especially Section 4.7 in [1]. The main difference to their paper is that they use a so-called Beckner-type inequality as their starting point (a device we will be able to recover from our results as well, see Corollary 3.4), while we are building on a modified Φ-Sobolev inequality.…”

Concentration inequalities on the multislice and for sampling without replacement

“…for any functions f, g on Ω κ (which is the Dirichlet form of the underlying Markov chain). Recently, in [APS20] it was shown that in the context of general Markov semigroups, Beckner inequalities with constants bounded away from zero as p ↓ 1 and modified log-Sobolev inequalities are equivalent. In their article, the authors provide numerous examples and applications, also briefly discussing the multislice.…”