Limit shape of random convex polygonal lines: Even more universality

TL;DR

This paper extends the universality of the limit shape of convex polygonal lines under various multiplicative measures, showing that a specific shape persists across a broad class of probability distributions, unlike the one-dimensional case.

Contribution

It broadens the class of measures for which the universal limit shape of convex polygonal lines is valid, including analogs of decomposable combinatorial structures.

Findings

Limit shape $oxed{ ext{the same shape}}$ holds for a wider class of measures.

Universality contrasts with one-dimensional case where limit shapes depend on distribution.

Explicit limit shape $oxed{ ext{the same shape}}$ confirmed for many measures.

Abstract

The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on $\mathbb{Z}_+^2$, starting at the origin and with the right endpoint $n=(n_1,n_2)\to\infty$. In the case of the uniform measure, an explicit limit shape $\gamma^*:=\{(x_1,x_2)\in\mathbb{R}_+^2\colon \sqrt{1-x_1}+\sqrt{x_2}=1\}$ was found independently by Vershik (1994), B\'ar\'any (1995), and Sinai (1994). Recently, Bogachev and Zarbaliev (2011) proved that the limit shape $\gamma^*$ is universal for a certain parametric family of multiplicative probability measures generalizing the uniform distribution. In the present work, the universality result is extended to a much wider class of multiplicative measures, including (but not limited to) analogs of the three meta-types of decomposable combinatorial structures -- multisets, selections and assemblies. This result is in sharp contrast with the one-dimensional case where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type.