Motivated by connections via the Knutson-Tao hive model, to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave (real-valued) functions on an equilateral lattice with periodic Hessians. After suitable transformations the question can be rephrased as one involving the scaling limit of random semiconcave functions on a discrete torus. The resulting set of semiconcave functions forms a convex polytope \(P_n(s)\) , where s parameterizes the extent of semiconcavity. The \(\ell _\infty \) diameter of \(P_n(s)\) is shown to be bounded below by \(c(s) n^2\) , where c(s) is a positive constant depending only on s. It has been shown in [18] that \(\sigma (s) = - \lim _{n\rightarrow \infty } \left( \frac{1}{n^2}\right) \log \textrm{vol} P_n(s)\) is well defined and convex; in fact that \(\exp (- \sigma (s))\) is concave. We use this surface tension to show that when s is such that a subgradient \(w = (w_0, w_1, w_2)\) of \(\sigma (s)\) belongs to the cone \(\begin{aligned} w_0^2 + w_1^2 + w_2^2 < 2\left( w_0 w_1 + w_1 w_2 + w_2 w_0\right) , \end{aligned}\) (which happens to be true for when \(s_0 = s_1 \le s_2\) ,) then for any \({\epsilon }> 0,\) \(\lim _{n \rightarrow \infty } \mathbb {P}\left[ \Vert g\Vert _\infty > n^{\frac{7}{4} + {\epsilon }}\right] = 0\) where g is sampled from the uniform measure on \(P_n(s)\) .
