Convergence of uniform triangulations under the Cardy embedding

TL;DR

This paper demonstrates that uniform triangulations embedded via the Cardy method converge to the Brownian disk in the scaling limit, connecting discrete percolation models with continuous Liouville quantum gravity surfaces.

Contribution

It proves joint convergence of metric and measures for uniform triangulations under the Cardy embedding to the Brownian disk, including new percolation scaling limit results.

Findings

Convergence of triangulation metrics and measures to the Brownian disk.

Scaling limit of critical percolation crossing probabilities.

Validation of the Cardy embedding as a discrete conformal approximation.

Abstract

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 $\sqrt{8/3}$-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.