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 κ′ ↓ 4 in the subcritical setting, the space-filling SLEκ′ in a disk degenerates to the CLE4 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 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) from this pair, such that the Brownian motion A encodes the LQG distance from the CLE loops to the boundary of the disk and the Brownian motion B encodes the boundary lengths of the CLE4 loops. In contrast to the subcritical setting, the pair (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 metric, and growth fragmentations.