Limit shape of random convex polygonal lines: Even more universality

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 = ( n 1 , n 2 ) → ∞ . In the case of the uniform measure, an explicit limit shape γ ⁎ : = { ( x 1 , x 2 ) ∈ R + 2 : 1 − x 1 + x 2 = 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.