# 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.

## 244 Citations

### Scaling limit of large triangulations of polygons

- Mathematics
- 2019

We prove that large random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge to the Brownian disk.

### Cut Vertices in Random Planar Maps

- MathematicsAofA
- 2020

It is shown that X_n/n → c in probability (for some explicit c>0) and for so-called subcritial subclasses of planar maps like outerplanar maps the authors obtain a central limit theorem, too.

### The scaling limit of random cubic planar graphs

- Mathematics
- 2022

— We study the random simple connected cubic planar graph Cn with an even number n of vertices. We show that the Brownian map arises as Gromov– Hausdorff–Prokhorov scaling limit of Cn as n ∈ 2N tends…

### On the Riemann surface type of Random Planar Maps

- Mathematics
- 2011

We show that the (random) Riemann surfaces of the Angel-Schramm Uniform Infinite Planar Triangulation and of Sheffield's infinite necklace construction are both parabolic. In other words, Brownian…

### Equivalence of zero entropy and the Liouville property for stationary random graphs

- Mathematics
- 2015

We prove that any stationary random graph satisfying a growth condition and having positive entropy almost surely admits an infinite dimensional space of bounded harmonic functions. Applications to…

### The Incipient Infinite Cluster of the Uniform Infinite Half-Planar Triangulation

- Mathematics
- 2017

We introduce the Incipient Infinite Cluster (IIC) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation (UIHPT), which is the local limit of large random…

### Planar stochastic hyperbolic infinite triangulations

- Mathematics
- 2014

Pursuing the approach of Angel & Ray, we introduce and study a family of random infinite triangulations of the full-plane that satisfy a natural spatial Markov property. These new random lattices…

### Every planar graph with the Liouville property is amenable

- MathematicsRandom Struct. Algorithms
- 2020

A strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, and it is obtained that every planar non-amenable graph admits Dirichlet harmonic functions.

### Probabilistic Aspects of Infinite Trees and Surfaces

- Mathematics
- 2003

The conditional probability measure on the set of trees containing a given finite tree is computed to determine the distribution of the number of vertices at a given distance from the root, and thereby the Hausdorff dimension associated with this measure.

### Local convergence of large random triangulations coupled with an Ising model

- MathematicsTransactions of the American Mathematical Society
- 2020

We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size
n
n
decorated by an Ising configuration with a weight proportional to the…

## References

SHOWING 1-10 OF 39 REFERENCES

### Largest 4-Connected Components of 3-Connected Planar Triangulations

- MathematicsRandom Struct. Algorithms
- 1995

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

- MathematicsEur. J. Comb.
- 1994

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.

### A Census of Planar Triangulations

- MathematicsCanadian 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

- MathematicsSIAM J. Discret. Math.
- 1999

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

- Mathematics
- 2002

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

- Mathematics
- 2000

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

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

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

- MathematicsCanadian 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…