Uniform Infinite Planar Triangulations

By Omer Angel, Oded Schramm. 2002

261 citations recorded by the source corpus.

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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Citations in the corpus

Local limit of labeled trees and expected volume growth in a random quadrangulation (2003, methodology): “To avoid this we make the following assumption (see also [8]):”
Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps (2014, background): “The distribution ζ̂ is itself the uniform ordering of a distribution μ̂ on (Z+)(4) which we defined by μ̂(k1, k2, k3, k4) = k1b1 + k2b2 + k3b3 + k4b4 bi μ(k1, k2, k3, k4) for i ∈ [4] and k1, k2, k3, k4 ∈ Z+ and where b1, b2, b3, b4 are some positive numbers which depend on q.”
Some families of increasing planar maps (2007, methodology): “We refer to Angel & Schramm [5], Chassaing & Schaeffer [13] Bouttier & al.”
Unimodular random trees (2012, background): “Such a limit of a sequence of finite networks is called the random weak limit, the local (weak) limit, the distributional limit, or the Benjamini–Schramm limit of the sequence.”
Local structure of random quadrangulations (2005, background): “Theorem 1 for triangulations was proved by Angel and Schramm [2].”
Random geometry on the sphere (2014, background): “1 as an infinite planar map, that is, with a given embedding in the sphere, and this makes it possible to give a stronger form of the local convergence (see [7] and [27]).”
Geodesics in large planar maps and in the Brownian map (2008, background): “The problem of establishing a convergence of the type (1) was raised by Schramm [36] in the setting of triangulations.”
Recurrence of planar graph limits (2012, background): “(See [21], [3], [15], [5] and the references within.”
Hyperbolic and Parabolic Unimodular Random Maps (2016, background): “When Gn is a uniformly chosen map of size n, this construction gives rise to the aforementioned UIPT and UIPQ.”
Limit of normalized quadrangulations: The Brownian map (2004, background): “Thefunction encodesa treeT (see Section 3.2)and f isa function com patiblewith :ifs and s0aresuch that (s;s0)= (s)= (s0),then s and s0 encode the sam e point ofT and the third condition ensuresthatf isa function ofthepointsofT ;thisproperty iscalled the snake property.”
Percolation on uniform infinite planar maps (2013, methodology): “This can be ensured by dealing only with rigid quadrangulations, as in [5]: we say that a rooted quadrangulation with holes q is rigid if no quadrangulation of the sphere includes two different copies of q with coinciding roots.”
The topological structure of scaling limits of large planar maps (2006, background): “See Bouttier’s thesis [7] and the references therein for applications of these bijections to the statistical physics of random surfaces.”