Log concavity and concentration of Lipschitz functions on the Boolean hypercube

TL;DR

This paper introduces a new notion of log-concavity for measures on the Boolean hypercube, proving they satisfy significant concentration inequalities for Lipschitz functions, extending known results from continuous to discrete settings.

Contribution

It defines $eta$-semi-log-concavity on the hypercube and proves these measures exhibit strong concentration bounds, including for measures with the Rayleigh property.

Findings

Measures with $eta$-semi-log-concavity satisfy variance bounds for Lipschitz functions.

Measures with the Rayleigh property exhibit nontrivial concentration.

Concentration bounds extend to measures under external fields with non-positive correlations.

Abstract

It is well-known that measures whose density is the form $e^{-V}$ where $V$ is a uniformly convex potential on $\RR^n$ attain strong concentration properties. In search of a notion of log-concavity on the discrete hypercube, we consider measures on $\{-1,1\}^n$ whose multi-linear extension $f$ satisfies $\log \nabla^2 f(x) \preceq \beta \Id$, for $\beta \geq 0$, which we refer to as $\beta$-semi-log-concave. We prove that these measures satisfy a nontrivial concentration bound, namely, any Hamming Lipchitz test function $\varphi$ satisfies $\Var_\nu[\varphi] \leq n^{2-C_\beta}$ for $C_\beta>0$. As a corollary, we prove a concentration bound for measures which exhibit the so-called Rayleigh property. Namely, we show that for measures such that under any external field (or exponential tilt), the correlation between any two coordinates is non-positive, Hamming-Lipschitz functions admit nontrivial concentration.