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Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces

Published version
Peer-reviewed

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Authors

Gwynne, EN 
Miller, Jason 

Abstract

In a recent series of works, Miller and Sheffield constructed a metric on 8/3-Liouville quantum gravity (LQG) under which 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 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 SLE8/3 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 SLE8/3. 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 SLE8/3.

Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with SLE8/3 on 8/3-LQG.

Description

Keywords

Metric gluing, Schramm-Loewner evolution, Brownian surfaces, Liouville quantum gravity

Journal Title

Annals of Probability

Conference Name

Journal ISSN

0091-1798
2168-894X

Volume Title

47

Publisher

Institute of Mathematical Statistics

Rights

All rights reserved