It is well-known that measures whose density is the form e(-V) where V is a uniformly convex potential on R-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 del(2)f (x)= 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 phi satisfies Var(nu)phi 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. (c) 2022 Elsevier Inc. All rights reserved.
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