Convergence of uniform triangulations under the Cardy embedding
METADATA ONLY
Loading...
Author / Producer
Date
2023
Publication Type
Journal Article
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
Abstract
We consider an embedding of planar maps into an equilateral triangle Δ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov’s proof of Cardy’s formula. Under the Cardy embedding, the planar map induces a metric and an area measure on Δ and a boundary measure on ∂Δ. We prove that for uniformly sampled triangulations, the metric and the measures converge jointly in the scaling limit to the Brownian disk conformally embedded into Δ (i.e., to the √(8/3)-Liouville quantum gravity disk). As part of our proof, we prove scaling limit results for critical site percolation on the uniform triangulations, in a quenched sense. In particular, we establish the scaling limit of the percolation crossing probability for a uniformly sampled triangulation with four boundary marked points.
Permanent link
Publication status
published
External links
Editor
Book title
Journal / series
Volume
230 (1)
Pages / Article No.
93 - 203
Publisher
International Press
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
09453 - Werner, Wendelin (ehemalig) / Werner, Wendelin (former)
Notes
Funding
Related publications and datasets
Is new version of: