Modified log-Sobolev inequalities and two-level concentration

Abstract

We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form Ent µ (e f ) ≤ ρ 2 E µ e f Γ(f ) 2 for some difference operator Γ, and show how it implies two-level concentration inequalities akin to the Hanson-Wright or Bernstein inequality. This can be applied to the continuous (e. g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, . . . ).Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group S n and for slices of the hypercube to prove Talagrand's convex distance inequality, and provide concentration inequalities for locally Lipschitz functions on S n . Some examples of known statistics are worked out, for which we obtain the correct order of fluctuations, which is consistent with central limit theorems.

Large deviations and a phase transition in the Block Spin Potts models

“…[18, Proposition 2.4, Theorem 1.5]. Moreover, (5.3) gives rise to subgaussian tail bounds for Lipschitztype functions f (in the sense of |df | ≤ 1) by applying the Herbst argument, see e. g. [35] (where also slightly more advanced situations are discussed, cf. Section 2.4).…”