Schramm-Loewner Evolution and Liouville Quantum Gravity

Schramm-Loewner Evolution and Liouville Quantum Gravity

We show that when two boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary length-preserving way) the resulting interface is a random curve called the Schramm-Loewner evolution. We also develop a theory of quantum fractal measures (consisten...

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Bibliographic Details
Main Authors: Duplantier, Bertrand (Author), Sheffield, Scott Roger (Contributor)
Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
Format: Article
Language:English
Published: American Physical Society (APS), 2012-01-27T19:42:54Z.
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Summary:We show that when two boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary length-preserving way) the resulting interface is a random curve called the Schramm-Loewner evolution. We also develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under conformal welding maps related to Schramm-Loewner evolution. As an application, we construct quantum length and boundary intersection measures on the Schramm-Loewner evolution curve itself.
French National Research Agency (ANR) (Grant No. ANR-08-BLAN-0311-CSD5)
Institut national des sciences de l'univers (France) (Grant No. CNRS-PEPS-PTI 2010)
National Science Foundation (U.S.) (Grant No. DMS 0403182/064558)
National Science Foundation (U.S.) (Grant No. OISE 0730136)

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