Convex chains in Z^2

Details Convex chains in Z^2 Convex chains in Z^2

2006-12-26

arxiv.org/abs/math/0612770v1

Mathematics

Probability

18 pages

Scientific paper

A detailed combinatorial analysis of lattice convex polygonal lines of N^2 joining 0 to (n,n) is presented. We derive consequences on the line having the largest number of vertices as well as the cardinal and limit shape of lines having few vertices. The proof refines a statistical physical method used by Sinai to obtain the typical behavior of these lines, allied to some Fourier analysis. Limit shapes of convex lines joining 0 to (n,n) and having a given total length are also characterized.

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