Random discrete concave functions on an equilateral lattice with periodic Hessians

TL;DR

This paper investigates the behavior of random discrete concave functions on an equilateral lattice with periodic Hessians, showing they concentrate around quadratic functions and establishing bounds on their size as the lattice size grows.

Contribution

It introduces a convex polytope of semiconcave functions with periodic Hessians and proves concentration and size bounds for functions sampled uniformly from this set.

Findings

Functions concentrate around quadratic functions as lattice size increases

The diameter of the convex polytope grows at least quadratically with lattice size

Probability of large deviations diminishes with increasing lattice size

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.