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 Z+2, starting at the origin and with the right endpoint n=(n1,n2)→∞. In the case of the uniform measure, an explicit limit shape γ⁎:={(x1,x2)∈R+2:1−x1+x2=1} was found independently by Vershik (1994) [19], Bárány (1995) [3], and Sinaĭ (1994) [16]. Recently, Bogachev and Zarbaliev (1999) [5] proved that the limit shape γ⁎ 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.