We show that a random concave function having a periodic Hessian of a fixed average value $s$ on an equilateral lattice concentrates around a quadratic function if $s$ satisfies certain conditions. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$, incur addition by a linear function (this condition is equivalent to the periodicity of the Hessian of $g$). We identify the functions in this set with a convex polytope $P_n(s)$, where $s$ corresponds to the average Hessian. We show that 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, if $s$ satisfies certain conditions, for any $\epsilon_0 > 0$, the normalized Lebesgue measure of all points in $P_n(s)$ that are not contained in a $n^2$ dimensional cube $Q$ of sidelength $ \epsilon_0n^2$, centered at the unique (up to addition of a linear term) quadratic polynomial with Hessian $s$, tends to $0$ as $n$ tends to $\infty$.