Convex chains in Z^2

Nathanael Enriquez (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: grand canonical ensemble; polylogarithm; limit shapes; local limit theorem.; local limit theorem

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Nathanael Enriquez. Convex chains in Z^2. 2006. ⟨hal-00122105v1⟩