Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense

TL;DR

This paper proves joint convergence of height functions for bipolar-oriented triangulations and their duals to correlated Brownian motions, confirming a conjecture and advancing the understanding of LQG surfaces decorated by dual SLE curves.

Contribution

It establishes the joint scaling limit of a bipolar-oriented triangulation and its dual, showing convergence to correlated Brownian motions encoding an LQG surface with dual SLE curves.

Findings

Joint convergence of height functions to correlated Brownian motions

Confirmation of a conjecture by Kenyon et al.

Foundation for connecting LQG with random permutons like the Baxter permuton.

Abstract

Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together with the mating-of-trees construction of Liouville quantum gravity (LQG) due to Duplantier, Miller, and Sheffield (2014), they proved that random bipolar-oriented planar maps converge in the scaling limit to a $\sqrt{4/3}$-LQG surface decorated by an independent SLE$_{12}$ in the peanosphere sense, meaning that the height functions of a particular pair of trees on the maps converge in the scaling limit to the correlated planar Brownian motion which encodes the SLE-decorated LQG surface. We improve this convergence result by proving that the pair of height functions for an infinite-volume random bipolar-oriented triangulation and the pair of height functions for its dual map converge jointly in law in the scaling limit to the two planar Brownian motions which encode the same $\sqrt{4/3}$-LQG surface decorated by both an SLE$_{12}$ curve and the ``dual'' SLE$_{12}$ curve which travels in a direction perpendicular (in the sense of imaginary geometry) to the original curve. This confirms a conjecture of Kenyon, Miller, Sheffield, and Wilson (2015). Our paper is the starting point of recent works connecting LQG and random permutons such as the Baxter permuton.