Abstract
A detailed combinatorial analysis of planar convex lattice polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained.
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D. M. Acketa and J. D. Žunić, On the maximal number of edges of convex digital polygons included into an m × m-grid, J. Combin. Theory Ser. A, 69 (1995), 358–368.
V. I. Arnold, Statistics of integral convex polygons, Funktsional. Anal. i Prilozhen., 14 (1980), 1–3.
R. Arratia and S. Tavaré, Independent process approximations for random combinatorial structures, Adv. Math., 104 (1994), 90–154.
I. Bárány, The limit shape of convex lattice polygons, Discrete Comput. Geom., 13 (1995), 279–295.
I. Bárány, Sylvester’s question: the probability that n points are in convex position, Ann. Probab., 27 (1999), 2020–2034.
I. Bárány, G. Rote, W. Steiger and C.-H. Zhang, A central limit theorem for convex chains in the square, Discrete Comput. Geom. 23 (2000), 35–50.
L. V. Bogachev and S. M. Zarbaliev, Universality of the limit shape of convex lattice polygonal lines, Ann. Probab., 39 (2011), 2271–2317.
E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337–357.
J. Bureaux, Partitions of large unbalanced bipartites, Math. Proc. Cambridge Philos. Soc., 157 (2014), 469–487.
J. Bureaux and N. Enriquez, On the number of lattice convex chains, Discrete Anal. (2016), Paper No. 19, 15 pp.
P. Erdős and J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J., 8 (1941), 335–345.
V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Math. Z. 24 (1926), 500–518.
W. M. Schmidt, Integer points on curves and surfaces, Monatsh. Math., 99 (1985), 45–72.
Y. G. Sinaĭ, A probabilistic approach to the analysis of the statistics of convex polygonal lines, Funktsional. Anal. i Prilozhen. 28 (1994), 41–48, 96.
P. Valtr, Probability that n random points are in convex position, Discrete Comput. Geom., 13 (1995), 637–643.
A. Vershik, The limit form of convex integral polygons and related problems, Funktsional. Anal. i Prilozhen., 28 (1994), 16–25, 95.
A. Vershik and Y. Yakubovich, The limit shape and fluctuations of random partitions of naturals with fixed number of summands, Mosc. Math. J., 1 (2001), 457–468, 472.
A. Vershik and O. Zeitouni, Large deviations in the geometry of convex lattice polygons, Israel J. Math., 109 (1999), 13–27.
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Bureaux, J., Enriquez, N. Asymptotics of convex lattice polygonal lines with a constrained number of vertices. Isr. J. Math. 222, 515–549 (2017). https://doi.org/10.1007/s11856-017-1599-3
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DOI: https://doi.org/10.1007/s11856-017-1599-3