  In a groundbreaking work, Duplantier, Miller and Sheffield showed that
subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner
evolutions (SLE) can be described by the mating of two continuum random trees.
In this paper, we consider the counterpart of their result for critical LQG and
SLE, i.e., for the case when $\gamma^2=\kappa=16/\kappa=4$. We prove that as
one sends $\kappa \downarrow 4$ in the subcritical setting, the space-filling
SLE$_\kappa$ in a disk degenerates to the CLE$_4$ exploration introduced by
Werner and Wu, along with a collection of i.i.d.\ coin tosses indexed by the
branch points of the exploration. Furthermore, in the
$\kappa=16/\gamma^2\downarrow 4$ limit, the pair of continuum random trees
collapse into a single continuum random tree, and we observe that upon applying
an appropriate affine transform to the encoding Brownian motions before taking
the limit, we get convergence to a pair of independent Brownian motions
$(A,B)$. The Brownian motion $A$ encodes the LQG distance from the CLE loops to
the boundary of the disk, while the Brownian motion $B$ encodes the boundary
lengths of the CLE$_4$ loops. In contrast to the subcritical setting, $(A,B)$
does not determine the CLE-decorated LQG surface.
