# Asymptotics of convex lattice polygonal lines with a constrained number of vertices

@article{Bureaux2016AsymptoticsOC, title={Asymptotics of convex lattice polygonal lines with a constrained number of vertices}, author={Julien Bureaux and Nathanael Enriquez}, journal={Israel Journal of Mathematics}, year={2016}, volume={222}, pages={515-549} }

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.

## References

SHOWING 1-10 OF 19 REFERENCES

### On the number of lattice convex chains

- Mathematics
- 2016

An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeroes of the…

### The limit shape of convex lattice polygons

- MathematicsDiscret. Comput. Geom.
- 1995

It is proved here that, asn→∞, almost all convex (1/n)ℤ2-lattice polygons lying in the square [−1, 1]2 are very close to a fixed convex set.

### Integer points on curves and surfaces

- Mathematics
- 1985

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ℝ3…

### Large deviations in the geometry of convex lattice polygons

- Mathematics
- 1999

We provide a full large deviation principle (LDP) for the uniform measure on certain ensembles of convex lattice polygons. This LDP provides for the analysis of concentration of the measure on convex…

### On the Maximal Number of Edges of Convex Digital Polygons Included into an m x m -Grid

- MathematicsJ. Comb. Theory, Ser. A
- 1995

### Sylvester's question : The probability that n points are in convex position

- Mathematics
- 1999

For a convex body K in the plane, let p(n, K) denote the probability that n random, independent, and uniform points from K are in convex position, that is, none of them lies in the convex hull of the…

### Partitions of large unbalanced bipartites

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2014

Abstract We compute the asymptotic behaviour of the number of partitions of large vectors (n1, n2) of ℤ+2 in the critical regime n1 ≍ √n2 and in the subcritical regime n1 = o(√n2). This work…

### A Central Limit Theorem for Convex Chains in the Square

- MathematicsDiscret. Comput. Geom.
- 2000

Under uniform probability, this work proves an almost sure limit theorem for these chains that uses only probabilistic arguments, and which strengthens similar limit shape statements established by other authors.

### Universality of the limit shape of convex lattice polygonal lines

- Mathematics
- 2008

Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as…

### Independent Process Approximations for Random Combinatorial Structures

- Mathematics
- 1994

Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted…