Log concavity and concentration of Lipschitz functions on the Boolean hypercube

Abstract

It is well-known that measures whose density is the form e−V where V is a uniformly convex potential on Rn 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⁡∇2f(x)⪯βIn, for β≥0, which we refer to as β-semi-log-concave. We prove that these measures satisfy a nontrivial concentration bound, namely, any Hamming Lipchitz test function φ satisfies Varν[φ]≤n2−Cβ for Cβ>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.