  We set the foundation for a series of works aimed at proving strong relations
between uniform random planar maps and Liouville quantum gravity (LQG). Our
method relies on a bijective encoding of site-percolated planar triangulations
by certain 2D lattice paths. Our bijection parallels in the discrete setting
the \emph{mating-of-trees} framework of LQG and Schramm-Loewner evolutions
(SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two
correspondences allows us to relate uniform site-percolated triangulations to
$\sqrt{8/3}$-LQG and SLE$_6$. In particular, we establish the convergence of
several functionals of the percolation model to continuous random objects
defined in terms of $\sqrt{8/3}$-LQG and SLE$_6$. For instance, we show that
the exploration tree of the percolation converges to a branching SLE$_6$, and
that the collection of percolation cycles converges to the conformal loop
ensemble CLE$_6$. We also prove convergence of counting measure on the pivotal
points of the percolation. Our results play an essential role in several other
works, including a program for showing convergence of the conformal structure
of uniform triangulations and works which study the behavior of random walk on
the uniform infinite planar triangulation.
