Uniform Infinite Planar Triangulations

Abstract

The existence of the weak limit as n --> infinity 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.

36 pages, 4 figures; Journal revised version

Cited by in corpus (41)

• The topological structure of scaling limits of large planar maps
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• Limit of normalized quadrangulations: The Brownian map
• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Quantum Loewner Evolution
• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• Random stable looptrees
• On the spectral dimension of causal triangulations
• Scaling limits of random planar maps with a unique large face
• Invariance principles for random bipartite planar maps
• Random walks on combs
• Unimodular Random Trees
• Percolation on uniform infinite planar maps
• Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs
• Recurrence of bipartite planar maps
• Local limits of uniform triangulations in high genus
• Uniform infinite planar triangulation and related time-reversed critical branching process
• The two uniform infinite quadrangulations of the plane have the same law
• Unimodular Hyperbolic Triangulations: Circle Packing and Random Walk
• A Boltzmann approach to percolation on random triangulations
• Stability of geodesics in the Brownian map
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• Local convergence for permutations and local limits for uniform $\rho$-avoiding permutations with $|\rho |=3$
• Collisions of Random Walks in Reversible Random Graphs
• Bounds on the critical line via transfer matrix methods for an Ising model coupled to causal dynamical triangulations
• Volumes in the Uniform Infinite Planar Triangulation: from skeletons to generating functions
• The skeleton of the UIPT, seen from infinity
• How fast planar maps get swallowed by a peeling process
• Growth of uniform infinite causal triangulations
• Spectral dimension of trees with a unique infinite spine
• Uniform infinite half-planar quadrangulations with skewness
• Planarity and non-separating cycles in uniform high genus quadrangulations
• Trees with exponential height dependent weight
• Random cubic planar graphs converge to the Brownian sphere
• Critical Ising model on random triangulations of the disk: enumeration and local limits
• Ising model on random triangulations of the disk: phase transition
• Generic Ising Trees
• Appearance of vertices of infinite order in a model of random trees
• Simple peeling of planar maps with application to site percolation
• The generating function of planar Eulerian orientations
• Trees with power-like height dependent weight