Inverse problem of the limit shape for convex lattice polygonal lines

TL;DR

This paper addresses the inverse problem of the limit shape for convex lattice polygonal lines, demonstrating that any smooth convex arc can be realized as a limit shape under an appropriate probability measure.

Contribution

It introduces a method to construct probability measures that produce any given smooth convex arc as the limit shape of convex lattice polygonal lines.

Findings

Any smooth convex arc can be realized as a limit shape.

Constructs probability measures for arbitrary convex curves.

Extends understanding of limit shape inverse problems.

Abstract

It is known that random convex polygonal lines on $\mathbb{Z}_+^2$ (with the endpoints fixed at $0=(0,0)$ and $n=(n_1,n_2)\to\infty$) have a limit shape with respect to the uniform probability measure, identified as the parabola arc $\sqrt{c\myp(1-x_1)}+\sqrt{x_2}=\sqrt{c}$, where $n_2/n_1\to c$. The present paper is concerned with the inverse problem of the limit shape. We show that for any strictly convex, $C^3$-smooth arc $\gamma\subset\mathbb{R}_+^2$ starting at the origin, there is a probability measure $P_n^\gamma$ on convex polygonal lines, under which the curve $\gamma$ is their limit shape.