Limit shape of random convex polygonal lines: Even more universality

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.