Distances on the $\mathrm{CLE}_4$, critical Liouville quantum gravity and $3/2$-stable maps

The purpose of this article is threefold. First, we show that when one explores a conformal loop ensemble of parameter $\kappa=4$ ($\mathrm{CLE}_4$) on an independent $2$-Liouville quantum gravity ($2$-LQG) disk, the surfaces which are cut out are independent quantum disks. To achieve this, we rely on approximations of the explorations of a $\mathrm{CLE}_4$: we first approximate the $\mathrm{SLE}_4^{\langle \mu \rangle}(-2)$ explorations for $\mu \in \mathbb{R}$ using explorations of the $\mathrm{CLE}_\kappa$ as $\kappa \uparrow 4$ and then we approximate the uniform exploration by letting $\mu \to \infty$. Second, we describe the relation between the so-called natural quantum distance and the conformally invariant distance to the boundary introduced by Werner and Wu. Third, we establish the scaling limit of the distances from the boundary to the large faces of $3/2$-stable maps and relate the limit to the $\mathrm{CLE}_4$-decorated $2$-LQG.