  Motivated by connections to random matrices, Littlewood-Richardson
coefficients and tilings, we study random discrete concave functions on an
equilateral lattice. We show that such functions having a periodic Hessian of a
fixed average value $- s$ concentrate around a quadratic function. We consider
the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that
when shifted by an element of $n \mathbb L$ have a periodic discrete Hessian,
with period $n \mathbb L$. We add a convex quadratic of Hessian $s$; the sum is
then periodic with period $n \mathbb L$, and view this as a mean zero function
$g$ on the set of vertices $V(\mathbb{T}_n)$ of a torus $\mathbb{T}_n :=
\frac{\mathbb{Z}}{n\mathbb{Z}}\times \frac{\mathbb{Z}}{n\mathbb{Z}}$ whose
Hessian is dominated by $s$. The resulting set of semiconcave functions forms a
convex polytope $P_n(s)$. The $\ell_\infty$ diameter of $P_n(s)$ is bounded
below by $c(s) n^2$, where $c(s)$ is a positive constant depending only on $s$.
Our main result is that under certain conditions, that are met for example when
$s_0 = s_1 \leq s_2$, for any $\epsilon > 0,$ we have $$\lim_{n \rightarrow 0}
\mathbb{P}\left[\|g\|_\infty > n^{\frac{7}{4} + \epsilon}\right] = 0$$ if $g$
is sampled from the uniform measure on $P_n(s)$. Each $g \in P_n(s)$
corresponds to a kind of honeycomb. We obtain concentration results for these
as well.
