  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.
