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

TL;DR

This paper establishes a correspondence between metric gluings of Liouville quantum gravity surfaces and their Brownian map counterparts, revealing new geometric structures and linking them to SLE curves.

Contribution

It identifies metric gluings of LQG surfaces with boundary, connecting them to Brownian surfaces and SLE curves, advancing the understanding of LQG's geometric and probabilistic properties.

Findings

Gluing two Brownian half-planes yields an LQG wedge with SLE$_{8/3}$

Boundary identification of Brownian half-planes produces LQG cones with SLE$_{8/3}$

Results connect LQG surface gluings with scaling limits of self-avoiding walks

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.