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UNIQUENESS OF THE WELDING PROBLEM FOR SLE AND LIOUVILLE QUANTUM GRAVITY

Part of: Geometric probability and stochastic geometry Special processes Markov processes

Published online by Cambridge University Press:  05 July 2019

Oliver McEnteggart ,
Jason Miller  and
Wei Qian Open the ORCID record for Wei Qian [Opens in a new window]
Oliver McEnteggart
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK (jpmiller@statslab.cam.ac.uk)
Jason Miller
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK (jpmiller@statslab.cam.ac.uk)
Wei Qian
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK (jpmiller@statslab.cam.ac.uk)
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Abstract

We give a simple set of geometric conditions on curves $\unicode[STIX]{x1D702}$, $\widetilde{\unicode[STIX]{x1D702}}$ in $\mathbf{H}$ from $0$ to $\infty$ so that if $\unicode[STIX]{x1D711}:\mathbf{H}\rightarrow \mathbf{H}$ is a homeomorphism which is conformal off $\unicode[STIX]{x1D702}$ with $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})=\widetilde{\unicode[STIX]{x1D702}}$ then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm–Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $\unicode[STIX]{x1D702}$ is a non-space-filling $\text{SLE}_{\unicode[STIX]{x1D705}}$ curve in $\mathbf{H}$ from $0$ to $\infty$, and $\unicode[STIX]{x1D711}$ is a homeomorphism which is conformal on $\mathbf{H}\setminus \unicode[STIX]{x1D702}$, and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})$, $\unicode[STIX]{x1D702}$ are equal in distribution, then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$. Applying this result for $\unicode[STIX]{x1D705}=4$ establishes that the welding operation for critical ($\unicode[STIX]{x1D6FE}=2$) LQG is well defined. Applying it for $\unicode[STIX]{x1D705}\in (4,8)$ gives a new proof that the welding of two independent $\unicode[STIX]{x1D705}/4$-stable looptrees of quantum disks to produce an $\text{SLE}_{\unicode[STIX]{x1D705}}$ on top of an independent $4/\sqrt{\unicode[STIX]{x1D705}}$-LQG surface is well defined.

Keywords

Conformal weldingLiouville quantum gravitySchramm–Loewner evolution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 757 - 783
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Copyright
© Cambridge University Press 2019

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