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Recent works have shown that random triangulations decorated by critical($p=1/2$) Bernoulli site percolation converge in the scaling limit to a$\sqrt{8/3}$-Liouville quantum gravity (LQG) surface (equivalently, a Browniansurface) decorated by SLE$_6$ in two different ways:  1. The triangulation, viewed as a curve-decorated metric measure spaceequipped with its graph distance, the counting measure on vertices, and asingle percolation interface converges with respect to a version of theGromov-Hausdorff topology.  2. There is a bijective encoding of the site-percolated triangulation bymeans of a two-dimensional random walk, and this walk converges to thecorrelated two-dimensional Brownian motion which encodes SLE$_6$-decorated$\sqrt{8/3}$-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield(2014); this is sometimes called $\textit{peanosphere convergence}$.  We prove that one in fact has $\textit{joint}$ convergence in both of thesetwo senses simultaneously. We also improve the metric convergence result byshowing that the map decorated by the full collection of percolation interfaces(rather than just a single interface) converges to $\sqrt{8/3}$-LQG decoratedby CLE$_6$ in the metric space sense.  This is the first work to prove simultaneous convergence of any random planarmap model in the metric and peanosphere senses. Moreover, this work is animportant step in an ongoing program to prove that random triangulationsembedded into $\mathbb C$ via the so-called $\textit{Cardy embedding}$ convergeto $\sqrt{8/3}$-LQG.

Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense