Brownian half-plane excursion and critical Liouville quantum gravity

In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be obtained by gluing together a pair of Brownian motions. In this paper, we study the counterpart of their result in the critical case via a limiting argument. In particular, we prove that as one sends kappa 'down arrow 4$\kappa {\prime } \downarrow 4$ in the subcritical setting, the space-filling SLE kappa '$_{\kappa {\prime }}$ in a disk degenerates to the CLE4$_4$ (where CLE is conformal loop ensembles) exploration introduced by Werner and Wu, along with a collection of independent and identically distributed coin tosses indexed by the branch points of the exploration. Furthermore, in the same limit, we observe that although the pair of initial Brownian motions collapses to a single one, one can still extract two different independent Brownian motions (A,B)$(A,B)$ from this pair, such that the Brownian motion A$A$ encodes the LQG distance from the CLE loops to the boundary of the disk and the Brownian motion B$B$ encodes the boundary lengths of the CLE4$_4$ loops. In contrast to the subcritical setting, the pair (A,B)$(A,B)$ does not determine the CLE-decorated LQG surface. Our paper also contains a discussion of relationships to random planar maps, the conformally invariant CLE4$_4$ metric and growth fragmentations.