Random discrete concave functions on an equilateral lattice with periodic Hessians

Abstract

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.