Acta Mathematica

Volume 230 (2023)

Number 1

Convergence of uniform triangulations under the Cardy embedding

Pages: 93 – 203



Nina Holden (Department of Mathematics, ETH Zürich, Switzerland)

Xin Sun (University of Pennsylvania, Philadelphia, Penn., U.S.A.)


We consider an embedding of planar maps into an equilateral triangle $\Delta$ 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 $\Delta$ and a boundary measure on $\partial \Delta$. 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 $\Delta$ (i.e., to the $\sqrt{\frac{8}{3}}$-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.

Received 22 March 2020

Received revised 17 April 2021

Accepted 1 June 2021

Published 24 March 2023