Inverse problem of the limit shape for convex lattice polygonal lines

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.

References in corpus (2)

• Universality of the limit shape of convex lattice polygonal lines
• Limit shape of random convex polygonal lines: Even more universality

Cited by in corpus (1)

• Limit shape of random convex polygonal lines: Even more universality