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July 2019 Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces
Ewain Gwynne, Jason Miller
Ann. Probab. 47(4): 2303-2358 (July 2019). DOI: 10.1214/18-AOP1309

Abstract

In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane and disk).

We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal $\mathrm{SLE}_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane $\mathrm{SLE}_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane $\mathrm{SLE}_{8/3}$.

Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with $\mathrm{SLE}_{8/3}$ on $\sqrt{8/3}$-LQG.

Citation

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Ewain Gwynne. Jason Miller. "Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces." Ann. Probab. 47 (4) 2303 - 2358, July 2019. https://doi.org/10.1214/18-AOP1309

Information

Received: 1 October 2016; Revised: 1 July 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114718
MathSciNet: MR3980922
Digital Object Identifier: 10.1214/18-AOP1309

Subjects:
Primary: 60D05, 60J67

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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