# Mating of trees for critical Liouville quantum gravity

Juhan Aru, Nina Holden, Ellen Powell, Xin Sun

Juhan Aru, Nina Holden, Ellen Powell, Xin Sun

**1/9/21**Published in : arXiv:2109.00275

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.