Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces
Authors
Gwynne, EN
Miller, Jason
Publication Date
2019Journal Title
Annals of Probability
ISSN
0091-1798
Publisher
Institute of Mathematical Statistics
Volume
47
Issue
4
Pages
2303--2358
Type
Article
This Version
VoR
Metadata
Show full item recordCitation
Gwynne, E., & Miller, J. (2019). Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces. Annals of Probability, 47 (4), 2303--2358. https://doi.org/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 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 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 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 SLE$_{8/3}$ on $\sqrt{8/3}$-LQG.
Keywords
Metric gluing, Schramm-Loewner evolution, Brownian surfaces, Liouville quantum gravity
Embargo Lift Date
2100-01-01
Identifiers
External DOI: https://doi.org/10.1214/18-AOP1309
This record's URL: https://www.repository.cam.ac.uk/handle/1810/287973
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