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Conformal weldings of random surfaces: SLE and the quantum gravity zipper

2016; Institute of Mathematical Statistics; Volume: 44; Issue: 5 Linguagem: Inglês

10.1214/15-aop1055

2168-894X

Scott Sheffield⋆,

Geometry and complex manifolds

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.