We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm–Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path-decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.
"Conformal weldings of random surfaces: SLE and the quantum gravity zipper." Ann. Probab. 44 (5) 3474 - 3545, September 2016. https://doi.org/10.1214/15-AOP1055