Limit shape of random convex polygonal lines on Z^2: Even more universality

Details Limit shape of random convex polygonal lines on Z^2: Even more universality Limit shape of random convex polygonal lines on Z^2: Even more universality

2011-11-15

arxiv.org/abs/1111.3529v1

Mathematics

Probability

32 pages

Scientific paper

The paper is concerned with the limit shape (under some probability measure) of convex polygonal lines on Z_+^2 starting at the origin and with the right endpoint n = (n_1,n_2) -> infinity. In the case of the uniform measure, the explicit limit shape \gamma* was found independently by Vershik, B\'ar\'any and Sinai. Bogachev and Zarbaliev recently showed that the limit shape \gamma* is universal in a certain class of measures analogous to multisets in the theory of decomposable combinatorial structures. In the present work, we extend the universality result to a much wider class of measures, including (but not limited to) analogues of 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.

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