Convergence of uniform triangulations under the Cardy embedding

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.