Limit shape of random convex polygonal lines: Even more universality

作者：Bogachev Leonid V^*

来源：Journal of Combinatorial Theory - Series A, 2014, 127: 353-399.

DOI：10.1016/j.jcta.2014.07.005

• Convex lattice polygonal line
• Limit shape
• Multiplicative measures
• Local limit theorem
• Mobius inversion formula
• Generating function
• Cumulants

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)) -%26gt;infinity. In the case of the uniform measure, an explicit limit shape gamma* := {(x(1), x(2)) is an element of R-+(2) : root 1-x(1) + root x(2) = 1} was found independently by Vershik (1994) [19], Barany (1995) [3], and Sinai (1994) [16]. Recently, Bogachev and Zarbaliev (1999) [5] 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 metatypes 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 depend on the distributional type.

• 出版日期2014-9