# Modified log-Sobolev inequalities and two-level concentration

## TL;DR

This paper establishes that modified log-Sobolev inequalities lead to two-level concentration inequalities, applicable in continuous and discrete settings, and demonstrates their use in proving Talagrand's inequality and analyzing fluctuations of statistics.

## Contribution

It introduces a general framework connecting modified log-Sobolev inequalities to two-level concentration and applies it to symmetric groups and hypercube slices.

## Key findings

- Derived two-level concentration inequalities from mLSI.
- Proved Talagrand's convex distance inequality using mLSI.
- Obtained fluctuation orders consistent with CLTs for known statistics.

## Abstract

We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_{\mu}(e^f) \le \tfrac{\rho}{2} \mathbb{E}_\mu e^f \Gamma(f)^2$ for some difference operator $\Gamma$, 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, \ldots). 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.