Random concave functions on an equilateral lattice with periodic Hessians I: entropy and Laplacians

TL;DR

This paper demonstrates that random concave functions with periodic Hessians on an equilateral lattice exhibit a quadratic scaling limit, with their measure concentrating around a specific quadratic polynomial as the lattice size grows.

Contribution

It establishes the quadratic scaling limit for such functions and characterizes the geometric structure of the set of these functions via convex polytopes.

Findings

Normalized measure concentrates near a quadratic polynomial

Diameter of the convex polytope scales as n^2

Measure outside a small cube tends to zero as n increases

Abstract

We show that a random concave function having a periodic hessian on an equilateral lattice has a quadratic scaling limit, if the average hessian of the function 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 this set, up to addition by a constant, 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, 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 $2 \epsilon_0 n^2$, centered at the unique (up to addition of a linear term) quadratic polynomial with hessian $s$, tends to $0$ as $n$ tends to $\infty$.