Mating of trees for critical Liouville quantum gravity

TL;DR

This paper extends the mating of trees framework to critical Liouville quantum gravity (LQG) and SLE, showing how subcritical structures degenerate and converge to new critical objects with Brownian motion encodings.

Contribution

It proves the critical case of LQG and SLE, demonstrating degeneration of subcritical structures and convergence to a single continuum random tree with Brownian motion encodings.

Findings

Subcritical SLE$_ppa$ degenerates to CLE$_4$ exploration.

Pair of continuum random trees collapses into a single tree at criticality.

Brownian motions encode distances and boundary lengths in the critical limit.

Abstract

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.