The SLE loop via conformal welding of quantum disks

TL;DR

This paper demonstrates that the SLE$_$ loop measure can be derived from conformal welding of Liouville quantum gravity disks, linking quantum surfaces with Schramm-Loewner evolution loops.

Contribution

It establishes a rigorous connection between SLE loops and conformal welding of LQG disks, advancing understanding of quantum surface interfaces.

Findings

SLE$_$ loops arise from conformal welding of LQG disks.

Scaling limits of decorated quadrangulations converge to LQG sphere with SLE$_{8/3}$.

The result supports the integrability of the conformal loop ensemble.

Abstract

We prove that the SLE$_\kappa$ loop measure arises naturally from the conformal welding of two $\gamma$-Liouville quantum gravity (LQG) disks for $\gamma^2 = \kappa \in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.