The Annals of Probability

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

Ewain Gwynne and Jason Miller

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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.

Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2303-2358.

Dates
Received: October 2016
Revised: July 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205710

Digital Object Identifier
doi:10.1214/18-AOP1309

Mathematical Reviews number (MathSciNet)
MR3980922

Zentralblatt MATH identifier
07114718

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

Keywords
Metric gluing Schramm–Loewner evolution Brownian surfaces Liouville quantum gravity

Citation

Gwynne, Ewain; Miller, Jason. Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces. Ann. Probab. 47 (2019), no. 4, 2303--2358. doi:10.1214/18-AOP1309. https://projecteuclid.org/euclid.aop/1562205710


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