The Annals of Probability

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

Scott Sheffield

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Abstract

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.

Article information

Source
Ann. Probab., Volume 44, Number 5 (2016), 3474-3545.

Dates
Received: October 2014
Revised: August 2015
First available in Project Euclid: 21 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1474462104

Digital Object Identifier
doi:10.1214/15-AOP1055

Mathematical Reviews number (MathSciNet)
MR3551203

Zentralblatt MATH identifier
06653523

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Conformal welding Gaussian free field SLE imaginary geometry

Citation

Sheffield, Scott. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab. 44 (2016), no. 5, 3474--3545. doi:10.1214/15-AOP1055. https://projecteuclid.org/euclid.aop/1474462104


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