Convergence of uniform triangulations under the Cardy embedding

Understand

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).