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Conformal weldings of random surfaces: SLE and the quantum gravity zipper. (English) Zbl 1388.60144
Summary: 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.

MSC:
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G15 Gaussian processes
60G60 Random fields
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