Uniform Infinite Planar Triangulations

@article{Angel2002UniformIP,
  title={Uniform Infinite Planar Triangulations},
  author={Omer Angel and Oded Schramm},
  journal={Communications in Mathematical Physics},
  year={2002},
  volume={241},
  pages={191-213}
}
The existence of the weak limit as n→∞ of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane. 
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244 Citations
Highly Influential Citations
40
Background Citations
117
Methods Citations
38
Results Citations
5

244 Citations

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References

SHOWING 1-10 OF 39 REFERENCES

Largest 4-Connected Components of 3-Connected Planar Triangulations

It follows that almost all 3-connected triangulations with n vertices have a cycle of length at least n/2 + o(n) so that the number of vertices in the largest 4-connected component of Tn is asymptotic to n/ 2 with probability tending to 1 as n ∞.

Root Vertex Valency Distributions of Rooted Maps and Rooted Triangulations

The asymptotic distributions of root vertex valencies for rooted maps and rooted triangulations on general surfaces are obtained and it is shown that the asymPTotic distributions are independent of the surface.

Random Triangulations of the Plane

A Census of Planar Triangulations

  • W. T. Tutte
  • Mathematics
    Canadian Journal of Mathematics
  • 1962
Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that…

The Size of the Largest Components in Random Planar Maps

This paper derives some general results about the size of the largest component and applies them to a variety of types of planar maps.

Growth and percolation on the uniform infinite planar triangulation

AbstractA construction as a growth process for sampling of the uniform in- finite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an…

Recurrence of Distributional Limits of Finite Planar Graphs

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional…

Almost All Maps Are Asymmetric

A simple proof of the fact that for a wide variety of classes of maps on surfaces, almost all maps in the class are asymmetric is given.

A Census of Hamiltonian Polygons

  • W. T. Tutte
  • Mathematics
    Canadian Journal of Mathematics
  • 1962
In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are…

Minimal dynamical triangulations of random surfaces