Convergence of uniform triangulations under the Cardy embedding

We consider an embedding of planar maps into equilateral triangle $\Delta$ which we call the Cardy embedding. The is a discrete approximation conformal map based on percolation observables that are used in Smirnov's proof Cardy's formula. Under embedding, induces metric and area measure boundary $\partial \Delta$. prove for uniformly sampled triangulations, measures converge jointly scaling limit to Brownian disk conformally embedded (i.e., $\sqrt{8/3}$-Liouville quantum gravity disk). As part our proof, results critical site uniform quenched sense. In particular, establish crossing probability triangulation with four marked points.