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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 concentrationproperties. 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 toas $\beta$-semi-log-concave. We prove that these measures satisfy a nontrivialconcentration bound, namely, any Hamming Lipchitz test function $\varphi$satisfies $\Var_\nu[\varphi] \leq n^{2-C_\beta}$ for $C_\beta>0$. As acorollary, we prove a concentration bound for measures which exhibit theso-called Rayleigh property. Namely, we show that for measures such that underany external field (or exponential tilt), the correlation between any twocoordinates is non-positive, Hamming-Lipschitz functions admit nontrivialconcentration.

Log concavity and concentration of Lipschitz functions on the Boolean hypercube